| L(s) = 1 | − 2.23·2-s + 3.01·4-s + 3.67·7-s − 2.27·8-s − 5.02·11-s − 2.67·13-s − 8.23·14-s − 0.936·16-s − 3.30·17-s + 2.88·19-s + 11.2·22-s − 6.01·23-s + 5.98·26-s + 11.0·28-s + 5.13·29-s − 31-s + 6.64·32-s + 7.40·34-s − 7.90·37-s − 6.45·38-s + 9.55·41-s + 5.24·43-s − 15.1·44-s + 13.4·46-s + 1.06·47-s + 6.50·49-s − 8.05·52-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s + 1.38·7-s − 0.804·8-s − 1.51·11-s − 0.741·13-s − 2.19·14-s − 0.234·16-s − 0.801·17-s + 0.660·19-s + 2.39·22-s − 1.25·23-s + 1.17·26-s + 2.09·28-s + 0.952·29-s − 0.179·31-s + 1.17·32-s + 1.26·34-s − 1.29·37-s − 1.04·38-s + 1.49·41-s + 0.800·43-s − 2.28·44-s + 1.98·46-s + 0.155·47-s + 0.929·49-s − 1.11·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 3.30T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 + 6.01T + 23T^{2} \) |
| 29 | \( 1 - 5.13T + 29T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 - 9.55T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 - 0.521T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 0.598T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 - 9.87T + 73T^{2} \) |
| 79 | \( 1 + 2.40T + 79T^{2} \) |
| 83 | \( 1 - 2.91T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.85529153513141862166257662691, −7.32321299396848242488835645803, −6.52544804208912103825912902231, −5.38712659270518943518484428251, −4.94849734500749250640423769485, −4.02262935439959320825249269756, −2.45626299943142558833625461898, −2.24391377521808077283016838036, −1.09169466202034581134060943923, 0,
1.09169466202034581134060943923, 2.24391377521808077283016838036, 2.45626299943142558833625461898, 4.02262935439959320825249269756, 4.94849734500749250640423769485, 5.38712659270518943518484428251, 6.52544804208912103825912902231, 7.32321299396848242488835645803, 7.85529153513141862166257662691
