| L(s) = 1 | + 2-s + 0.435·3-s + 4-s − 0.796·5-s + 0.435·6-s + 3.50·7-s + 8-s − 2.80·9-s − 0.796·10-s + 3.32·11-s + 0.435·12-s − 5.74·13-s + 3.50·14-s − 0.347·15-s + 16-s − 5.02·17-s − 2.80·18-s − 2.58·19-s − 0.796·20-s + 1.52·21-s + 3.32·22-s + 23-s + 0.435·24-s − 4.36·25-s − 5.74·26-s − 2.53·27-s + 3.50·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.251·3-s + 0.5·4-s − 0.356·5-s + 0.177·6-s + 1.32·7-s + 0.353·8-s − 0.936·9-s − 0.251·10-s + 1.00·11-s + 0.125·12-s − 1.59·13-s + 0.936·14-s − 0.0896·15-s + 0.250·16-s − 1.21·17-s − 0.662·18-s − 0.593·19-s − 0.178·20-s + 0.333·21-s + 0.708·22-s + 0.208·23-s + 0.0889·24-s − 0.873·25-s − 1.12·26-s − 0.487·27-s + 0.662·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 0.435T + 3T^{2} \) |
| 5 | \( 1 + 0.796T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 + 5.74T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 5.64T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 + 8.67T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 7.02T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 + 8.50T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 - 0.567T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 - 0.978T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 8.30T + 97T^{2} \) |
| 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
−7.80781094375672816945232204160, −6.85796694242724249412227587438, −6.42962188823135223833090625053, −5.24109753080629051171113886715, −4.87524251835616260043988989346, −4.15766609073898745447341180248, −3.31234625769227148239672581037, −2.29267812084981612686586638458, −1.73568512348176238208564960499, 0,
1.73568512348176238208564960499, 2.29267812084981612686586638458, 3.31234625769227148239672581037, 4.15766609073898745447341180248, 4.87524251835616260043988989346, 5.24109753080629051171113886715, 6.42962188823135223833090625053, 6.85796694242724249412227587438, 7.80781094375672816945232204160
