L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 4·11-s + 2·13-s + 4·15-s − 2·17-s − 25-s + 4·27-s + 8·29-s − 8·31-s − 8·33-s + 4·37-s − 4·39-s − 10·41-s + 2·43-s − 2·45-s − 7·49-s + 4·51-s − 8·53-s − 8·55-s − 14·59-s − 4·65-s − 8·67-s − 8·71-s + 6·73-s + 2·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.03·15-s − 0.485·17-s − 1/5·25-s + 0.769·27-s + 1.48·29-s − 1.43·31-s − 1.39·33-s + 0.657·37-s − 0.640·39-s − 1.56·41-s + 0.304·43-s − 0.298·45-s − 49-s + 0.560·51-s − 1.09·53-s − 1.07·55-s − 1.82·59-s − 0.496·65-s − 0.977·67-s − 0.949·71-s + 0.702·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.219912223805299557784301757018, −8.496257439560674365393906531662, −7.54711260990795019997477179156, −6.55781115986726442238332444842, −6.13005225970784078836460370399, −4.96292524216913801622183832736, −4.20038931564431759564820391710, −3.24187553090721475339129588440, −1.41696183137349126770930412017, 0,
1.41696183137349126770930412017, 3.24187553090721475339129588440, 4.20038931564431759564820391710, 4.96292524216913801622183832736, 6.13005225970784078836460370399, 6.55781115986726442238332444842, 7.54711260990795019997477179156, 8.496257439560674365393906531662, 9.219912223805299557784301757018