L(s) = 1 | + 0.906·2-s − 1.17·4-s + 2.59·7-s − 2.88·8-s − 0.741·11-s − 3.78·13-s + 2.35·14-s − 0.258·16-s + 3.16·17-s − 19-s − 0.672·22-s − 0.570·23-s − 3.43·26-s − 3.05·28-s − 6·29-s + 5.83·31-s + 5.52·32-s + 2.87·34-s − 1.40·37-s − 0.906·38-s + 3.83·41-s − 2.59·43-s + 0.872·44-s − 0.517·46-s − 5.08·47-s − 0.258·49-s + 4.46·52-s + ⋯ |
L(s) = 1 | + 0.641·2-s − 0.588·4-s + 0.981·7-s − 1.01·8-s − 0.223·11-s − 1.05·13-s + 0.629·14-s − 0.0647·16-s + 0.768·17-s − 0.229·19-s − 0.143·22-s − 0.119·23-s − 0.673·26-s − 0.577·28-s − 1.11·29-s + 1.04·31-s + 0.977·32-s + 0.492·34-s − 0.230·37-s − 0.147·38-s + 0.599·41-s − 0.395·43-s + 0.131·44-s − 0.0763·46-s − 0.741·47-s − 0.0369·49-s + 0.618·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 0.906T + 2T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 + 0.741T + 11T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 23 | \( 1 + 0.570T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 - 0.160T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 - 4.19T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 3.78T + 97T^{2} \) |
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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
−7.87607745392649852985620323957, −7.52477694227810615553587234449, −6.33232238136744738867102887005, −5.62008645730983472565392969261, −4.85899745658348104876768509492, −4.50646757123219558250114641473, −3.47870499295558170559892349945, −2.63095027650360229120305264862, −1.47420594182419079080223822244, 0,
1.47420594182419079080223822244, 2.63095027650360229120305264862, 3.47870499295558170559892349945, 4.50646757123219558250114641473, 4.85899745658348104876768509492, 5.62008645730983472565392969261, 6.33232238136744738867102887005, 7.52477694227810615553587234449, 7.87607745392649852985620323957