| L(s) = 1 | − 2-s + 4-s − 0.414·5-s − 8-s + 0.414·10-s − 11-s + 1.17·13-s + 16-s − 2.17·17-s − 0.828·19-s − 0.414·20-s + 22-s + 3.24·23-s − 4.82·25-s − 1.17·26-s − 2.82·29-s + 6.48·31-s − 32-s + 2.17·34-s + 9.65·37-s + 0.828·38-s + 0.414·40-s + 4.65·41-s − 2.82·43-s − 44-s − 3.24·46-s − 9.24·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.185·5-s − 0.353·8-s + 0.130·10-s − 0.301·11-s + 0.324·13-s + 0.250·16-s − 0.526·17-s − 0.190·19-s − 0.0926·20-s + 0.213·22-s + 0.676·23-s − 0.965·25-s − 0.229·26-s − 0.525·29-s + 1.16·31-s − 0.176·32-s + 0.372·34-s + 1.58·37-s + 0.134·38-s + 0.0654·40-s + 0.727·41-s − 0.431·43-s − 0.150·44-s − 0.478·46-s − 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.128386901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128386901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 0.414T + 5T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 9.24T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 - 4.75T + 79T^{2} \) |
| 83 | \( 1 + 9.82T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.71830294762776455579518831457, −7.19430754090099976546372763906, −6.28709005722480928707401040083, −5.92361095041523925319748316832, −4.86308031073735847406379559640, −4.21959379999979578918023023464, −3.26188180739200364696851324383, −2.51978016143408214864950207166, −1.62165681714596334888669739203, −0.56690169144027953126232582918,
0.56690169144027953126232582918, 1.62165681714596334888669739203, 2.51978016143408214864950207166, 3.26188180739200364696851324383, 4.21959379999979578918023023464, 4.86308031073735847406379559640, 5.92361095041523925319748316832, 6.28709005722480928707401040083, 7.19430754090099976546372763906, 7.71830294762776455579518831457
