| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 4·11-s + 12-s − 13-s − 16-s − 2·17-s + 18-s − 4·19-s + 4·22-s − 8·23-s + 3·24-s − 26-s − 27-s − 2·29-s − 8·31-s + 5·32-s − 4·33-s − 2·34-s − 36-s − 6·37-s − 4·38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.277·13-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.852·22-s − 1.66·23-s + 0.612·24-s − 0.196·26-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.883·32-s − 0.696·33-s − 0.342·34-s − 1/6·36-s − 0.986·37-s − 0.648·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.491737413211789719412400168942, −8.951416944724915251595281792366, −7.895595238939559990092068768675, −6.66896007363382389039301575094, −6.08889791321956520018719790461, −5.15062543822582910502220466883, −4.22742899625398303254389449046, −3.60701057572819650856324347980, −1.90856486468770625999139740684, 0,
1.90856486468770625999139740684, 3.60701057572819650856324347980, 4.22742899625398303254389449046, 5.15062543822582910502220466883, 6.08889791321956520018719790461, 6.66896007363382389039301575094, 7.895595238939559990092068768675, 8.951416944724915251595281792366, 9.491737413211789719412400168942
