| L(s) = 1 | − 2·4-s − 5.25·7-s + 2.45·13-s + 4·16-s − 5.89·19-s − 5·25-s + 10.5·28-s + 11.1·31-s + 5.84·37-s + 9.98·43-s + 20.5·49-s − 4.90·52-s + 12.2·61-s − 8·64-s − 16.0·67-s + 0.0544·73-s + 11.7·76-s − 17.6·79-s − 12.8·91-s + 8.85·97-s + 10·100-s + 9.47·103-s − 6.27·109-s − 21.0·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.98·7-s + 0.680·13-s + 16-s − 1.35·19-s − 25-s + 1.98·28-s + 1.99·31-s + 0.960·37-s + 1.52·43-s + 2.93·49-s − 0.680·52-s + 1.56·61-s − 64-s − 1.95·67-s + 0.00636·73-s + 1.35·76-s − 1.98·79-s − 1.35·91-s + 0.898·97-s + 100-s + 0.933·103-s − 0.601·109-s − 1.98·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 5.25T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.45T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 - 11.1T + 31T^{2} \) |
| 37 | \( 1 - 5.84T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 16.0T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 0.0544T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−7.59628699300520965649180800692, −6.63460108567893945924794668194, −6.11499566415912530504803472664, −5.67591118116113773684498184987, −4.37379820046605395706154618229, −4.06444897393581446474919392252, −3.20849347332780585268741605872, −2.47341979823624841074061692368, −0.932877276837166783780316361544, 0,
0.932877276837166783780316361544, 2.47341979823624841074061692368, 3.20849347332780585268741605872, 4.06444897393581446474919392252, 4.37379820046605395706154618229, 5.67591118116113773684498184987, 6.11499566415912530504803472664, 6.63460108567893945924794668194, 7.59628699300520965649180800692
