| L(s) = 1 | + 3-s − 4·5-s − 3·7-s − 2·9-s − 2·11-s − 13-s − 4·15-s + 3·17-s + 19-s − 3·21-s + 23-s + 11·25-s − 5·27-s − 5·29-s + 8·31-s − 2·33-s + 12·35-s − 2·37-s − 39-s − 8·41-s − 4·43-s + 8·45-s − 8·47-s + 2·49-s + 3·51-s − 53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 1.13·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s − 1.03·15-s + 0.727·17-s + 0.229·19-s − 0.654·21-s + 0.208·23-s + 11/5·25-s − 0.962·27-s − 0.928·29-s + 1.43·31-s − 0.348·33-s + 2.02·35-s − 0.328·37-s − 0.160·39-s − 1.24·41-s − 0.609·43-s + 1.19·45-s − 1.16·47-s + 2/7·49-s + 0.420·51-s − 0.137·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−11.43965443835611430533032930003, −10.30984639175148445762941404826, −9.250246741517628766073180669878, −8.199448456904276296960401284970, −7.65960330166497750263390530877, −6.52280948162697403172717458153, −5.01088818916226849614299890353, −3.57195691104337905967362169037, −3.01072892067414194047812488684, 0,
3.01072892067414194047812488684, 3.57195691104337905967362169037, 5.01088818916226849614299890353, 6.52280948162697403172717458153, 7.65960330166497750263390530877, 8.199448456904276296960401284970, 9.250246741517628766073180669878, 10.30984639175148445762941404826, 11.43965443835611430533032930003
