L(s) = 1 | − 2-s − 3-s + 4-s − 0.924·5-s + 6-s − 4.24·7-s − 8-s + 9-s + 0.924·10-s + 2.47·11-s − 12-s − 13-s + 4.24·14-s + 0.924·15-s + 16-s + 0.549·17-s − 18-s − 4.66·19-s − 0.924·20-s + 4.24·21-s − 2.47·22-s − 2.37·23-s + 24-s − 4.14·25-s + 26-s − 27-s − 4.24·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.413·5-s + 0.408·6-s − 1.60·7-s − 0.353·8-s + 0.333·9-s + 0.292·10-s + 0.747·11-s − 0.288·12-s − 0.277·13-s + 1.13·14-s + 0.238·15-s + 0.250·16-s + 0.133·17-s − 0.235·18-s − 1.07·19-s − 0.206·20-s + 0.926·21-s − 0.528·22-s − 0.496·23-s + 0.204·24-s − 0.828·25-s + 0.196·26-s − 0.192·27-s − 0.802·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3380783898\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3380783898\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 0.924T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 17 | \( 1 - 0.549T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 - 7.33T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 + 0.968T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 3.20T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 + 0.591T + 89T^{2} \) |
| 97 | \( 1 + 7.00T + 97T^{2} \) |
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
−7.920299718318549130803023757255, −6.91429891986695972684143783286, −6.48388353554722804659395055787, −6.20275379041155116478387495456, −5.09484195171155125403135167785, −4.17271823130166573759609578952, −3.47738058979860054269760541404, −2.66673557425111615718295259501, −1.53721710403238630585466723072, −0.33103669072318425855869482144,
0.33103669072318425855869482144, 1.53721710403238630585466723072, 2.66673557425111615718295259501, 3.47738058979860054269760541404, 4.17271823130166573759609578952, 5.09484195171155125403135167785, 6.20275379041155116478387495456, 6.48388353554722804659395055787, 6.91429891986695972684143783286, 7.920299718318549130803023757255