| L(s) = 1 | + 2-s + 3-s + 4-s − 2.21·5-s + 6-s − 1.29·7-s + 8-s + 9-s − 2.21·10-s − 11-s + 12-s − 1.33·13-s − 1.29·14-s − 2.21·15-s + 16-s + 4.16·17-s + 18-s + 3.24·19-s − 2.21·20-s − 1.29·21-s − 22-s − 8.08·23-s + 24-s − 0.0791·25-s − 1.33·26-s + 27-s − 1.29·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.992·5-s + 0.408·6-s − 0.490·7-s + 0.353·8-s + 0.333·9-s − 0.701·10-s − 0.301·11-s + 0.288·12-s − 0.369·13-s − 0.346·14-s − 0.572·15-s + 0.250·16-s + 1.00·17-s + 0.235·18-s + 0.743·19-s − 0.496·20-s − 0.283·21-s − 0.213·22-s − 1.68·23-s + 0.204·24-s − 0.0158·25-s − 0.261·26-s + 0.192·27-s − 0.245·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 + 1.29T + 7T^{2} \) |
| 13 | \( 1 + 1.33T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 + 8.08T + 23T^{2} \) |
| 29 | \( 1 + 5.42T + 29T^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 - 2.69T + 59T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 - 1.68T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 - 6.20T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
| 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
−7.82402071686515276992097619978, −7.51029948395120363764308895078, −6.69450354627546287475369846115, −5.69467984474894698434513285171, −5.07214645097961746030464434782, −3.86630835987166975103028435457, −3.67862220389413578861427490759, −2.75097080050230370483001723298, −1.67657419920329934636464807088, 0,
1.67657419920329934636464807088, 2.75097080050230370483001723298, 3.67862220389413578861427490759, 3.86630835987166975103028435457, 5.07214645097961746030464434782, 5.69467984474894698434513285171, 6.69450354627546287475369846115, 7.51029948395120363764308895078, 7.82402071686515276992097619978
