| L(s) = 1 | − 2·5-s − 7-s − 11-s − 13-s + 2·17-s − 2·19-s + 5·23-s + 3·25-s + 29-s − 5·31-s + 2·35-s + 12·37-s − 12·41-s − 2·43-s − 13·47-s − 5·49-s − 3·53-s + 2·55-s + 10·59-s + 14·61-s + 2·65-s − 10·67-s − 3·71-s + 18·73-s + 77-s − 6·79-s − 20·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s + 0.485·17-s − 0.458·19-s + 1.04·23-s + 3/5·25-s + 0.185·29-s − 0.898·31-s + 0.338·35-s + 1.97·37-s − 1.87·41-s − 0.304·43-s − 1.89·47-s − 5/7·49-s − 0.412·53-s + 0.269·55-s + 1.30·59-s + 1.79·61-s + 0.248·65-s − 1.22·67-s − 0.356·71-s + 2.10·73-s + 0.113·77-s − 0.675·79-s − 2.19·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 50 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 13 T + 128 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 136 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 134 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82611697234271688414412595362, −7.50898527330744420624393846697, −7.04519982983922336884590464379, −6.94891149484792694563476116208, −6.35584935732090709988481210316, −6.32775347861085791542154207149, −5.54650581109369900733334901687, −5.39372717826936084295244431695, −4.83843424307612515696610206885, −4.73358007255185879330918658859, −4.18150046095782096695890874471, −3.70925907377571546642092449445, −3.40378584062483038159521330067, −3.13581233220736650668723410526, −2.44796049774225103585717184209, −2.31870065022993275870369587816, −1.32253232827672230913440058775, −1.16154507981986099226853740702, 0, 0,
1.16154507981986099226853740702, 1.32253232827672230913440058775, 2.31870065022993275870369587816, 2.44796049774225103585717184209, 3.13581233220736650668723410526, 3.40378584062483038159521330067, 3.70925907377571546642092449445, 4.18150046095782096695890874471, 4.73358007255185879330918658859, 4.83843424307612515696610206885, 5.39372717826936084295244431695, 5.54650581109369900733334901687, 6.32775347861085791542154207149, 6.35584935732090709988481210316, 6.94891149484792694563476116208, 7.04519982983922336884590464379, 7.50898527330744420624393846697, 7.82611697234271688414412595362
