L(s) = 1 | − 1.19·2-s − 1.42·3-s − 0.566·4-s + 1.70·6-s + 1.72·7-s + 3.07·8-s − 0.973·9-s − 6.17·11-s + 0.806·12-s − 2.79·13-s − 2.06·14-s − 2.54·16-s − 1.90·17-s + 1.16·18-s + 1.36·19-s − 2.45·21-s + 7.39·22-s − 1.31·23-s − 4.37·24-s + 3.34·26-s + 5.65·27-s − 0.977·28-s + 6.68·29-s + 9.60·31-s − 3.09·32-s + 8.78·33-s + 2.27·34-s + ⋯ |
L(s) = 1 | − 0.846·2-s − 0.821·3-s − 0.283·4-s + 0.695·6-s + 0.652·7-s + 1.08·8-s − 0.324·9-s − 1.86·11-s + 0.232·12-s − 0.775·13-s − 0.552·14-s − 0.636·16-s − 0.461·17-s + 0.274·18-s + 0.313·19-s − 0.535·21-s + 1.57·22-s − 0.274·23-s − 0.892·24-s + 0.656·26-s + 1.08·27-s − 0.184·28-s + 1.24·29-s + 1.72·31-s − 0.547·32-s + 1.52·33-s + 0.390·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 + 1.42T + 3T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 1.31T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 9.60T + 31T^{2} \) |
| 37 | \( 1 + 2.69T + 37T^{2} \) |
| 41 | \( 1 - 0.155T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 - 0.0487T + 47T^{2} \) |
| 53 | \( 1 - 6.75T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 8.73T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 + 7.62T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 0.436T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.945519500951555117659418325457, −7.21987004784238398694623293111, −6.34194042247590478481484284899, −5.38269234028942744093128052883, −4.93100580398900612775315590311, −4.46024808789393961003141216944, −2.96372637209096629529811230626, −2.19218843044338781419420591098, −0.876386315022375924962971823603, 0,
0.876386315022375924962971823603, 2.19218843044338781419420591098, 2.96372637209096629529811230626, 4.46024808789393961003141216944, 4.93100580398900612775315590311, 5.38269234028942744093128052883, 6.34194042247590478481484284899, 7.21987004784238398694623293111, 7.945519500951555117659418325457