L(s) = 1 | + 1.77·2-s + 3-s + 1.15·4-s − 1.23·5-s + 1.77·6-s + 2.47·7-s − 1.50·8-s + 9-s − 2.19·10-s + 2.85·11-s + 1.15·12-s − 5.44·13-s + 4.39·14-s − 1.23·15-s − 4.97·16-s + 17-s + 1.77·18-s − 3.45·19-s − 1.42·20-s + 2.47·21-s + 5.06·22-s + 3.71·23-s − 1.50·24-s − 3.47·25-s − 9.66·26-s + 27-s + 2.84·28-s + ⋯ |
L(s) = 1 | + 1.25·2-s + 0.577·3-s + 0.575·4-s − 0.552·5-s + 0.724·6-s + 0.935·7-s − 0.532·8-s + 0.333·9-s − 0.692·10-s + 0.860·11-s + 0.332·12-s − 1.50·13-s + 1.17·14-s − 0.318·15-s − 1.24·16-s + 0.242·17-s + 0.418·18-s − 0.791·19-s − 0.317·20-s + 0.539·21-s + 1.07·22-s + 0.775·23-s − 0.307·24-s − 0.695·25-s − 1.89·26-s + 0.192·27-s + 0.538·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.47T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 5.44T + 13T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 + 8.35T + 37T^{2} \) |
| 41 | \( 1 + 1.76T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 - 2.23T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.41T + 61T^{2} \) |
| 67 | \( 1 - 7.95T + 67T^{2} \) |
| 71 | \( 1 + 1.12T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.26841013014818725445477830716, −6.92790979331776444907740023231, −5.91128000555178396108523855105, −5.01622385631469376000569934794, −4.75845517078392161814563996093, −3.83417012922689566112391019097, −3.46899368118097952852145555065, −2.38915737193250805206384993711, −1.69263686110657156789871314717, 0,
1.69263686110657156789871314717, 2.38915737193250805206384993711, 3.46899368118097952852145555065, 3.83417012922689566112391019097, 4.75845517078392161814563996093, 5.01622385631469376000569934794, 5.91128000555178396108523855105, 6.92790979331776444907740023231, 7.26841013014818725445477830716