| L(s) = 1 | − 8·9-s + 2·13-s + 4·17-s − 6·23-s + 21·25-s + 4·27-s − 2·29-s − 6·43-s − 4·49-s + 22·53-s + 8·61-s − 26·79-s + 20·81-s + 28·101-s + 32·107-s + 50·113-s − 16·117-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 8/3·9-s + 0.554·13-s + 0.970·17-s − 1.25·23-s + 21/5·25-s + 0.769·27-s − 0.371·29-s − 0.914·43-s − 4/7·49-s + 3.02·53-s + 1.02·61-s − 2.92·79-s + 20/9·81-s + 2.78·101-s + 3.09·107-s + 4.70·113-s − 1.47·117-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 2.58·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.704873873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704873873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 - 2 T + 20 T^{2} - 94 T^{3} + 278 T^{4} - 94 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 4 T^{2} - 2 T^{3} + 14 T^{4} - 2 p T^{5} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 21 T^{2} + 249 T^{4} - 2006 T^{6} + 11626 T^{8} - 2006 p^{2} T^{10} + 249 p^{4} T^{12} - 21 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 52 T^{2} + 1368 T^{4} - 24280 T^{6} + 312502 T^{8} - 24280 p^{2} T^{10} + 1368 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 T + 16 T^{2} - 24 T^{3} + 10 T^{4} - 24 p T^{5} + 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 81 T^{2} + 3529 T^{4} - 102890 T^{6} + 2243374 T^{8} - 102890 p^{2} T^{10} + 3529 p^{4} T^{12} - 81 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 3 T + 79 T^{2} + 160 T^{3} + 2540 T^{4} + 160 p T^{5} + 79 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + T + 33 T^{2} + 2 T^{3} + 1414 T^{4} + 2 p T^{5} + 33 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 125 T^{2} + 8493 T^{4} - 384510 T^{6} + 13439406 T^{8} - 384510 p^{2} T^{10} + 8493 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 84 T^{2} + 2216 T^{4} + 61384 T^{6} - 5827706 T^{8} + 61384 p^{2} T^{10} + 2216 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 172 T^{2} + 16612 T^{4} - 1090964 T^{6} + 51790710 T^{8} - 1090964 p^{2} T^{10} + 16612 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 3 T + 87 T^{2} + 20 T^{3} + 4016 T^{4} + 20 p T^{5} + 87 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 137 T^{2} + 13681 T^{4} - 938098 T^{6} + 50458758 T^{8} - 938098 p^{2} T^{10} + 13681 p^{4} T^{12} - 137 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 11 T + 209 T^{2} - 1686 T^{3} + 16482 T^{4} - 1686 p T^{5} + 209 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 292 T^{2} + 39492 T^{4} - 3405100 T^{6} + 222712566 T^{8} - 3405100 p^{2} T^{10} + 39492 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 4 T + 132 T^{2} - 1068 T^{3} + 8646 T^{4} - 1068 p T^{5} + 132 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 168 T^{2} + 18332 T^{4} - 1592024 T^{6} + 121444582 T^{8} - 1592024 p^{2} T^{10} + 18332 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 160 T^{2} + 10920 T^{4} - 867532 T^{6} + 79016406 T^{8} - 867532 p^{2} T^{10} + 10920 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 305 T^{2} + 45549 T^{4} - 4497786 T^{6} + 354140538 T^{8} - 4497786 p^{2} T^{10} + 45549 p^{4} T^{12} - 305 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 13 T + 219 T^{2} + 2704 T^{3} + 22084 T^{4} + 2704 p T^{5} + 219 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 117 T^{2} + 15309 T^{4} - 1167406 T^{6} + 97224854 T^{8} - 1167406 p^{2} T^{10} + 15309 p^{4} T^{12} - 117 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 273 T^{2} + 54405 T^{4} - 7250626 T^{6} + 745491194 T^{8} - 7250626 p^{2} T^{10} + 54405 p^{4} T^{12} - 273 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 329 T^{2} + 69397 T^{4} - 9804594 T^{6} + 1094792986 T^{8} - 9804594 p^{2} T^{10} + 69397 p^{4} T^{12} - 329 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.21405734015989968020498965472, −4.98440302468318998975147661312, −4.86702135915485274525280771268, −4.74948158399407131467257916491, −4.51152952753275296891634453633, −4.26784484580085579827597102224, −4.20225991335644861338350385094, −4.07138265373757040033604461826, −3.96246563199485283872832726593, −3.69510754426125027902779695606, −3.30094798102949744681697092291, −3.18587997293312915614847256182, −3.17219161471432442018915212458, −3.15389631666158188227376774881, −2.98085641253099215978232509524, −2.97194403364512641709329826613, −2.32737899060447752938044896195, −2.29646857424947691540914743192, −2.21973291127902648990791518750, −2.09990632730266695058190433681, −1.70228935853159926908517975543, −1.11497915987280702210636644404, −1.03260316333940635867674308952, −0.973328140680099961676589984813, −0.32130902749978451068776818392,
0.32130902749978451068776818392, 0.973328140680099961676589984813, 1.03260316333940635867674308952, 1.11497915987280702210636644404, 1.70228935853159926908517975543, 2.09990632730266695058190433681, 2.21973291127902648990791518750, 2.29646857424947691540914743192, 2.32737899060447752938044896195, 2.97194403364512641709329826613, 2.98085641253099215978232509524, 3.15389631666158188227376774881, 3.17219161471432442018915212458, 3.18587997293312915614847256182, 3.30094798102949744681697092291, 3.69510754426125027902779695606, 3.96246563199485283872832726593, 4.07138265373757040033604461826, 4.20225991335644861338350385094, 4.26784484580085579827597102224, 4.51152952753275296891634453633, 4.74948158399407131467257916491, 4.86702135915485274525280771268, 4.98440302468318998975147661312, 5.21405734015989968020498965472
Plot not available for L-functions of degree greater than 10.
