| L(s) = 1 | + 2-s − 1.92·3-s + 4-s − 3.54·5-s − 1.92·6-s + 8-s + 0.691·9-s − 3.54·10-s + 0.289·11-s − 1.92·12-s − 3.97·13-s + 6.81·15-s + 16-s − 1.25·17-s + 0.691·18-s − 0.558·19-s − 3.54·20-s + 0.289·22-s + 3.97·23-s − 1.92·24-s + 7.59·25-s − 3.97·26-s + 4.43·27-s + 1.01·29-s + 6.81·30-s + 3.76·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.10·3-s + 0.5·4-s − 1.58·5-s − 0.784·6-s + 0.353·8-s + 0.230·9-s − 1.12·10-s + 0.0873·11-s − 0.554·12-s − 1.10·13-s + 1.76·15-s + 0.250·16-s − 0.304·17-s + 0.163·18-s − 0.128·19-s − 0.793·20-s + 0.0617·22-s + 0.828·23-s − 0.392·24-s + 1.51·25-s − 0.779·26-s + 0.853·27-s + 0.189·29-s + 1.24·30-s + 0.675·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.92T + 3T^{2} \) |
| 5 | \( 1 + 3.54T + 5T^{2} \) |
| 11 | \( 1 - 0.289T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 + 1.25T + 17T^{2} \) |
| 19 | \( 1 + 0.558T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 - 3.76T + 31T^{2} \) |
| 37 | \( 1 - 0.304T + 37T^{2} \) |
| 41 | \( 1 - 9.70T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 0.613T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.28T + 73T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.46810233739487387329760862748, −6.70574216240190817393986557498, −6.14020446897430087019181094679, −5.23354079956328045004895379162, −4.64801817179769248388140855107, −4.23004199335988415139574566442, −3.23661348003812990109732053985, −2.52191496765151793113060360591, −0.965285610790150693037070374632, 0,
0.965285610790150693037070374632, 2.52191496765151793113060360591, 3.23661348003812990109732053985, 4.23004199335988415139574566442, 4.64801817179769248388140855107, 5.23354079956328045004895379162, 6.14020446897430087019181094679, 6.70574216240190817393986557498, 7.46810233739487387329760862748
