L(s) = 1 | + 2-s − 2.98·3-s + 4-s − 1.98·5-s − 2.98·6-s − 7-s + 8-s + 5.89·9-s − 1.98·10-s − 5.55·11-s − 2.98·12-s − 6.48·13-s − 14-s + 5.91·15-s + 16-s + 3.37·17-s + 5.89·18-s + 5.27·19-s − 1.98·20-s + 2.98·21-s − 5.55·22-s + 6.73·23-s − 2.98·24-s − 1.06·25-s − 6.48·26-s − 8.61·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.72·3-s + 0.5·4-s − 0.887·5-s − 1.21·6-s − 0.377·7-s + 0.353·8-s + 1.96·9-s − 0.627·10-s − 1.67·11-s − 0.860·12-s − 1.79·13-s − 0.267·14-s + 1.52·15-s + 0.250·16-s + 0.819·17-s + 1.38·18-s + 1.21·19-s − 0.443·20-s + 0.650·21-s − 1.18·22-s + 1.40·23-s − 0.608·24-s − 0.213·25-s − 1.27·26-s − 1.65·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 + 6.48T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 - 3.74T + 37T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 - 2.39T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 + 3.25T + 53T^{2} \) |
| 59 | \( 1 + 4.64T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 8.64T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 7.42T + 97T^{2} \) |
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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
−7.50118813611607160805805216206, −7.06418161106290050198449724369, −6.04139350181836903596450787290, −5.46769668345681944470112283325, −4.84090437867839132781153770862, −4.54247929194674575770644014482, −3.23386920260253512886166278640, −2.60574400680031703126907573010, −0.951426305797181985413685011165, 0,
0.951426305797181985413685011165, 2.60574400680031703126907573010, 3.23386920260253512886166278640, 4.54247929194674575770644014482, 4.84090437867839132781153770862, 5.46769668345681944470112283325, 6.04139350181836903596450787290, 7.06418161106290050198449724369, 7.50118813611607160805805216206