| L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s + 9-s − 11-s − 2·12-s − 4·13-s − 15-s + 4·16-s + 3·17-s − 19-s + 2·20-s + 21-s − 3·23-s + 25-s + 27-s − 2·28-s − 9·29-s − 10·31-s − 33-s − 35-s − 2·36-s − 4·37-s − 4·39-s + 6·41-s − 43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s − 0.258·15-s + 16-s + 0.727·17-s − 0.229·19-s + 0.447·20-s + 0.218·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 1.67·29-s − 1.79·31-s − 0.174·33-s − 0.169·35-s − 1/3·36-s − 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.349195453153448745160274929435, −8.596981698368816702652082295587, −7.70622145793886853818069395455, −7.36355126599825039995964536459, −5.75320816325402874836433408548, −4.97316229451495260526011948731, −4.07791353239332549512579074638, −3.24782327620140806802994281062, −1.83055928022744764575111536586, 0,
1.83055928022744764575111536586, 3.24782327620140806802994281062, 4.07791353239332549512579074638, 4.97316229451495260526011948731, 5.75320816325402874836433408548, 7.36355126599825039995964536459, 7.70622145793886853818069395455, 8.596981698368816702652082295587, 9.349195453153448745160274929435
