L(s) = 1 | + 2-s − 2.37·3-s + 4-s − 2.07·5-s − 2.37·6-s − 0.108·7-s + 8-s + 2.64·9-s − 2.07·10-s − 2.80·11-s − 2.37·12-s + 0.0828·13-s − 0.108·14-s + 4.93·15-s + 16-s − 0.866·17-s + 2.64·18-s + 2.95·19-s − 2.07·20-s + 0.257·21-s − 2.80·22-s − 23-s − 2.37·24-s − 0.686·25-s + 0.0828·26-s + 0.834·27-s − 0.108·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.37·3-s + 0.5·4-s − 0.928·5-s − 0.970·6-s − 0.0408·7-s + 0.353·8-s + 0.882·9-s − 0.656·10-s − 0.845·11-s − 0.686·12-s + 0.0229·13-s − 0.0289·14-s + 1.27·15-s + 0.250·16-s − 0.210·17-s + 0.624·18-s + 0.677·19-s − 0.464·20-s + 0.0561·21-s − 0.598·22-s − 0.208·23-s − 0.485·24-s − 0.137·25-s + 0.0162·26-s + 0.160·27-s − 0.0204·28-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 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 + 0.108T + 7T^{2} \) |
| 11 | \( 1 + 2.80T + 11T^{2} \) |
| 13 | \( 1 - 0.0828T + 13T^{2} \) |
| 17 | \( 1 + 0.866T + 17T^{2} \) |
| 19 | \( 1 - 2.95T + 19T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 - 5.25T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 5.53T + 67T^{2} \) |
| 71 | \( 1 + 3.89T + 71T^{2} \) |
| 73 | \( 1 + 9.27T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 5.48T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68705578344328183862158024565, −6.84257155843597397040128407108, −6.07846482760223864546532059808, −5.64755147366990097011886596243, −4.69877037704394451091583030944, −4.42897471877567395836975045376, −3.35594992862249814660446995865, −2.52623036831299463284025410069, −1.07573711363988427018462096632, 0,
1.07573711363988427018462096632, 2.52623036831299463284025410069, 3.35594992862249814660446995865, 4.42897471877567395836975045376, 4.69877037704394451091583030944, 5.64755147366990097011886596243, 6.07846482760223864546532059808, 6.84257155843597397040128407108, 7.68705578344328183862158024565