| L(s) = 1 | + 2·4-s − 4·5-s − 2·9-s + 4·11-s + 16-s + 24·19-s − 8·20-s + 8·25-s − 24·29-s − 16·31-s − 4·36-s + 4·41-s + 8·44-s + 8·45-s − 8·49-s − 16·55-s + 4·59-s − 8·61-s − 2·64-s + 48·76-s − 4·80-s + 9·81-s − 64·89-s − 96·95-s − 8·99-s + 16·100-s + 32·101-s + ⋯ |
| L(s) = 1 | + 4-s − 1.78·5-s − 2/3·9-s + 1.20·11-s + 1/4·16-s + 5.50·19-s − 1.78·20-s + 8/5·25-s − 4.45·29-s − 2.87·31-s − 2/3·36-s + 0.624·41-s + 1.20·44-s + 1.19·45-s − 8/7·49-s − 2.15·55-s + 0.520·59-s − 1.02·61-s − 1/4·64-s + 5.50·76-s − 0.447·80-s + 81-s − 6.78·89-s − 9.84·95-s − 0.804·99-s + 8/5·100-s + 3.18·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7208687355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7208687355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 + 4 T + 8 T^{2} - 8 T^{3} - 41 T^{4} - 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 7 | \( 1 + 8 T^{2} + 46 T^{4} - 640 T^{6} - 5213 T^{8} - 640 p^{2} T^{10} + 46 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 2 T - 13 T^{2} + 10 T^{3} + 124 T^{4} + 10 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 20 T^{2} + 231 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 18 T^{2} + 563 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 36 T^{2} + 298 T^{4} - 2160 T^{6} + 30579 T^{8} - 2160 p^{2} T^{10} + 298 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 80 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 110 T^{2} + 139 p T^{4} + 266750 T^{6} + 11699428 T^{8} + 266750 p^{2} T^{10} + 139 p^{5} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 168 T^{2} + 16846 T^{4} + 1169280 T^{6} + 62232387 T^{8} + 1169280 p^{2} T^{10} + 16846 p^{4} T^{12} + 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 128 T^{2} + 8850 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 2 T + 35 T^{2} + 298 T^{3} - 2756 T^{4} + 298 p T^{5} + 35 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 104 T^{2} - 8 T^{3} + 9703 T^{4} - 8 p T^{5} - 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 158 T^{2} + 163 p T^{4} + 800270 T^{6} + 64991332 T^{8} + 800270 p^{2} T^{10} + 163 p^{5} T^{12} + 158 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 50 T^{2} + 1683 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 104 T^{2} + 4575 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 25 T^{2} - 8784 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−7.03359102898768566334571652894, −6.56722115902533300852606747793, −6.41039935120279862995092961411, −6.05615186042056899621944335188, −5.86777420491565887386810951752, −5.69972505269856589164084678426, −5.66794010593206933314305594383, −5.64602718897363475251973089842, −5.55387967066356602848588482080, −4.91569702126479824903541430576, −4.90869621782238256009088608540, −4.75567310202299895305231768175, −4.71292075716634527105090508002, −3.96963944904665922993769664424, −3.96718856080354666891603288247, −3.87973663604563963223802631918, −3.49130015533201372340507188095, −3.40143802318796413810831655341, −3.25308499998496659481242528402, −3.13343893345047280814276023466, −2.88985220502493558871282011110, −2.22143501198063388053769767323, −1.93537116038684257520250565973, −1.66426832167945821948089823812, −1.13154269407539701542357772075,
1.13154269407539701542357772075, 1.66426832167945821948089823812, 1.93537116038684257520250565973, 2.22143501198063388053769767323, 2.88985220502493558871282011110, 3.13343893345047280814276023466, 3.25308499998496659481242528402, 3.40143802318796413810831655341, 3.49130015533201372340507188095, 3.87973663604563963223802631918, 3.96718856080354666891603288247, 3.96963944904665922993769664424, 4.71292075716634527105090508002, 4.75567310202299895305231768175, 4.90869621782238256009088608540, 4.91569702126479824903541430576, 5.55387967066356602848588482080, 5.64602718897363475251973089842, 5.66794010593206933314305594383, 5.69972505269856589164084678426, 5.86777420491565887386810951752, 6.05615186042056899621944335188, 6.41039935120279862995092961411, 6.56722115902533300852606747793, 7.03359102898768566334571652894
Plot not available for L-functions of degree greater than 10.
