| L(s) = 1 | − 2·3-s − 4·5-s + 3·9-s + 4·11-s − 8·13-s + 8·15-s + 4·17-s − 4·23-s + 4·25-s − 4·27-s + 8·29-s − 8·31-s − 8·33-s + 8·37-s + 16·39-s + 4·41-s − 12·45-s − 8·51-s + 4·53-s − 16·55-s + 8·59-s − 16·61-s + 32·65-s + 8·69-s − 4·71-s + 8·73-s − 8·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s + 1.20·11-s − 2.21·13-s + 2.06·15-s + 0.970·17-s − 0.834·23-s + 4/5·25-s − 0.769·27-s + 1.48·29-s − 1.43·31-s − 1.39·33-s + 1.31·37-s + 2.56·39-s + 0.624·41-s − 1.78·45-s − 1.12·51-s + 0.549·53-s − 2.15·55-s + 1.04·59-s − 2.04·61-s + 3.96·65-s + 0.963·69-s − 0.474·71-s + 0.936·73-s − 0.923·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.44580358884736283410009563597, −7.37926055663323740349445626231, −6.74289247590207632968240427264, −6.72179868179206086224145312982, −6.12698372246636125989589359703, −5.87595854259471568082728533687, −5.38897811734563400694683718008, −5.15438545557841502301353931146, −4.59039069376921842145555086532, −4.47709280091476783126997238757, −3.99191189864677436049537304978, −3.94490002001419325246999053548, −3.21949935623726062744663923486, −3.07834121522320874497585029493, −2.17419224357951494962394437670, −2.12944986438998220534557064232, −1.10897008304573366247725079464, −0.925649264892962053697896084650, 0, 0,
0.925649264892962053697896084650, 1.10897008304573366247725079464, 2.12944986438998220534557064232, 2.17419224357951494962394437670, 3.07834121522320874497585029493, 3.21949935623726062744663923486, 3.94490002001419325246999053548, 3.99191189864677436049537304978, 4.47709280091476783126997238757, 4.59039069376921842145555086532, 5.15438545557841502301353931146, 5.38897811734563400694683718008, 5.87595854259471568082728533687, 6.12698372246636125989589359703, 6.72179868179206086224145312982, 6.74289247590207632968240427264, 7.37926055663323740349445626231, 7.44580358884736283410009563597
