| L(s) = 1 | − 2-s + 4-s − 1.47·5-s + 3.20·7-s − 8-s + 1.47·10-s + 0.0552·11-s − 0.805·13-s − 3.20·14-s + 16-s + 2.05·17-s + 3.67·19-s − 1.47·20-s − 0.0552·22-s − 2.81·25-s + 0.805·26-s + 3.20·28-s − 7.34·29-s − 7.95·31-s − 32-s − 2.05·34-s − 4.73·35-s + 3.08·37-s − 3.67·38-s + 1.47·40-s + 1.45·41-s + 12.5·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.661·5-s + 1.21·7-s − 0.353·8-s + 0.467·10-s + 0.0166·11-s − 0.223·13-s − 0.856·14-s + 0.250·16-s + 0.499·17-s + 0.843·19-s − 0.330·20-s − 0.0117·22-s − 0.562·25-s + 0.158·26-s + 0.605·28-s − 1.36·29-s − 1.42·31-s − 0.176·32-s − 0.353·34-s − 0.800·35-s + 0.507·37-s − 0.596·38-s + 0.233·40-s + 0.227·41-s + 1.91·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 0.0552T + 11T^{2} \) |
| 13 | \( 1 + 0.805T + 13T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 - 3.08T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 1.27T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 5.36T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.52T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 9.86T + 73T^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 8.25T + 89T^{2} \) |
| 97 | \( 1 + 7.01T + 97T^{2} \) |
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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
−7.57924502585192423353767354386, −7.07482668687193005563466786197, −5.84397577449614478824180659475, −5.47012267329206477005852275100, −4.49965992194647599791260264974, −3.83993891266461238603917901593, −2.94853616086692601302623598346, −1.93430312335768911750720636634, −1.22035675696718519935120637470, 0,
1.22035675696718519935120637470, 1.93430312335768911750720636634, 2.94853616086692601302623598346, 3.83993891266461238603917901593, 4.49965992194647599791260264974, 5.47012267329206477005852275100, 5.84397577449614478824180659475, 7.07482668687193005563466786197, 7.57924502585192423353767354386
