| L(s) = 1 | − 2·5-s + 2·7-s + 4·13-s − 4·17-s + 3·25-s + 4·29-s + 8·31-s − 4·35-s + 12·37-s − 4·41-s − 8·43-s + 16·47-s + 3·49-s − 4·53-s + 8·59-s + 12·61-s − 8·65-s − 8·67-s − 8·71-s + 12·73-s + 8·83-s + 8·85-s + 28·89-s + 8·91-s + 12·97-s − 12·101-s − 16·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 1.10·13-s − 0.970·17-s + 3/5·25-s + 0.742·29-s + 1.43·31-s − 0.676·35-s + 1.97·37-s − 0.624·41-s − 1.21·43-s + 2.33·47-s + 3/7·49-s − 0.549·53-s + 1.04·59-s + 1.53·61-s − 0.992·65-s − 0.977·67-s − 0.949·71-s + 1.40·73-s + 0.878·83-s + 0.867·85-s + 2.96·89-s + 0.838·91-s + 1.21·97-s − 1.19·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.901345972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901345972\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.756727455977410268972210879500, −8.719705295066796590010324459653, −8.393499457682629763557493487576, −8.002971567402841306859474131988, −7.66285984359465831336505803864, −7.28416789533615624429928783557, −6.62804375688143931796439452740, −6.62253044099171401687101428330, −5.98738154203801305752689045656, −5.70640050978020046432600140386, −4.98902846563598847837500199267, −4.73133848626764174879982771492, −4.17068702885430910471767880845, −4.16233469489290248844182120842, −3.32111187748106030094283408738, −3.11027494967774674529223194206, −2.25113899160505517991951338115, −2.03132498007677724299276864886, −0.924747989442578615768519181329, −0.77008678676796498021355722506,
0.77008678676796498021355722506, 0.924747989442578615768519181329, 2.03132498007677724299276864886, 2.25113899160505517991951338115, 3.11027494967774674529223194206, 3.32111187748106030094283408738, 4.16233469489290248844182120842, 4.17068702885430910471767880845, 4.73133848626764174879982771492, 4.98902846563598847837500199267, 5.70640050978020046432600140386, 5.98738154203801305752689045656, 6.62253044099171401687101428330, 6.62804375688143931796439452740, 7.28416789533615624429928783557, 7.66285984359465831336505803864, 8.002971567402841306859474131988, 8.393499457682629763557493487576, 8.719705295066796590010324459653, 8.756727455977410268972210879500
