| L(s) = 1 | + 6·9-s − 4·11-s + 12·19-s − 32·29-s + 8·31-s + 20·41-s + 4·49-s + 8·59-s + 24·61-s − 16·71-s − 40·79-s + 17·81-s − 36·89-s − 24·99-s − 8·101-s − 24·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + ⋯ |
| L(s) = 1 | + 2·9-s − 1.20·11-s + 2.75·19-s − 5.94·29-s + 1.43·31-s + 3.12·41-s + 4/7·49-s + 1.04·59-s + 3.07·61-s − 1.89·71-s − 4.50·79-s + 17/9·81-s − 3.81·89-s − 2.41·99-s − 0.796·101-s − 2.29·109-s + 2/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.624330469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624330469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 579 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - 68 T^{2} + 2086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 9939 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 258 T^{2} + 27011 T^{4} - 258 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 250 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 278 T^{2} + 32451 T^{4} - 278 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 251 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1286 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.88196006610654043788924276460, −5.78644788559578376695634494450, −5.72188514815253540085561803480, −5.46276845800538196616417743550, −5.40503923442936961687093705651, −5.26518934330761625167021583001, −5.15535957666994943401976620610, −4.45511239226416405361604708017, −4.45501572105016041421808354421, −4.26989138065251087423972225246, −4.13789733979090084155441852924, −3.86700038999519217131011326156, −3.73192579380079466697086674021, −3.55123709431033132021329775657, −2.94637977471264276048510161505, −2.90848804964381052085099897870, −2.88772420141092971452159805080, −2.45468226332452055816394000756, −2.12253965181989150401535365916, −1.89836500422513399073781545863, −1.61467278539112314076277788082, −1.27138145235014972874011789197, −1.16807230865654579689269821329, −0.75165534813702181885774385676, −0.18764947194953940710212597448,
0.18764947194953940710212597448, 0.75165534813702181885774385676, 1.16807230865654579689269821329, 1.27138145235014972874011789197, 1.61467278539112314076277788082, 1.89836500422513399073781545863, 2.12253965181989150401535365916, 2.45468226332452055816394000756, 2.88772420141092971452159805080, 2.90848804964381052085099897870, 2.94637977471264276048510161505, 3.55123709431033132021329775657, 3.73192579380079466697086674021, 3.86700038999519217131011326156, 4.13789733979090084155441852924, 4.26989138065251087423972225246, 4.45501572105016041421808354421, 4.45511239226416405361604708017, 5.15535957666994943401976620610, 5.26518934330761625167021583001, 5.40503923442936961687093705651, 5.46276845800538196616417743550, 5.72188514815253540085561803480, 5.78644788559578376695634494450, 5.88196006610654043788924276460
