| L(s) = 1 | − 2-s + 0.234·3-s + 4-s − 3.98·5-s − 0.234·6-s + 1.98·7-s − 8-s − 2.94·9-s + 3.98·10-s − 0.00620·11-s + 0.234·12-s − 1.24·13-s − 1.98·14-s − 0.935·15-s + 16-s − 1.24·17-s + 2.94·18-s + 19-s − 3.98·20-s + 0.466·21-s + 0.00620·22-s + 3.24·23-s − 0.234·24-s + 10.8·25-s + 1.24·26-s − 1.39·27-s + 1.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.135·3-s + 0.5·4-s − 1.78·5-s − 0.0958·6-s + 0.750·7-s − 0.353·8-s − 0.981·9-s + 1.26·10-s − 0.00187·11-s + 0.0677·12-s − 0.345·13-s − 0.530·14-s − 0.241·15-s + 0.250·16-s − 0.302·17-s + 0.694·18-s + 0.229·19-s − 0.891·20-s + 0.101·21-s + 0.00132·22-s + 0.676·23-s − 0.0479·24-s + 2.17·25-s + 0.244·26-s − 0.268·27-s + 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.234T + 3T^{2} \) |
| 5 | \( 1 + 3.98T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 + 0.00620T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 - 9.98T + 29T^{2} \) |
| 31 | \( 1 + 8.91T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 + 0.105T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 0.871T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 9.06T + 73T^{2} \) |
| 79 | \( 1 + 9.98T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
| 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
−7.57752543636796531558185257511, −7.13703658298063746281255858443, −6.35082003684254290020856636507, −5.20021080893410333511049384503, −4.74169998644604278529399418409, −3.72105927135769347713619534156, −3.13935420282744679616145547832, −2.23087505405738314908744275625, −0.952772224387019133994294308913, 0,
0.952772224387019133994294308913, 2.23087505405738314908744275625, 3.13935420282744679616145547832, 3.72105927135769347713619534156, 4.74169998644604278529399418409, 5.20021080893410333511049384503, 6.35082003684254290020856636507, 7.13703658298063746281255858443, 7.57752543636796531558185257511
