L(s) = 1 | + 0.447·2-s − 2.52·3-s − 1.80·4-s − 1.13·6-s + 3.76·7-s − 1.69·8-s + 3.40·9-s + 4.61·11-s + 4.55·12-s + 3.87·13-s + 1.68·14-s + 2.84·16-s − 3.87·17-s + 1.52·18-s − 5.29·19-s − 9.53·21-s + 2.06·22-s − 9.04·23-s + 4.29·24-s + 1.73·26-s − 1.01·27-s − 6.78·28-s − 4.06·29-s + 3.52·31-s + 4.66·32-s − 11.6·33-s − 1.73·34-s + ⋯ |
L(s) = 1 | + 0.316·2-s − 1.46·3-s − 0.900·4-s − 0.461·6-s + 1.42·7-s − 0.600·8-s + 1.13·9-s + 1.39·11-s + 1.31·12-s + 1.07·13-s + 0.450·14-s + 0.710·16-s − 0.940·17-s + 0.358·18-s − 1.21·19-s − 2.08·21-s + 0.440·22-s − 1.88·23-s + 0.877·24-s + 0.339·26-s − 0.195·27-s − 1.28·28-s − 0.754·29-s + 0.632·31-s + 0.825·32-s − 2.03·33-s − 0.297·34-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 | \( 1 - 0.447T + 2T^{2} \) |
| 3 | \( 1 + 2.52T + 3T^{2} \) |
| 7 | \( 1 - 3.76T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 9.04T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 + 8.42T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 7.38T + 43T^{2} \) |
| 47 | \( 1 - 5.04T + 47T^{2} \) |
| 53 | \( 1 + 3.84T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 + 7.15T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 7.83T + 73T^{2} \) |
| 79 | \( 1 - 2.00T + 79T^{2} \) |
| 83 | \( 1 + 3.77T + 83T^{2} \) |
| 89 | \( 1 + 0.414T + 89T^{2} \) |
| 97 | \( 1 + 1.63T + 97T^{2} \) |
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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
−7.77969292181990310865578301524, −6.61535538693502859985042344789, −6.19016337796064430599673156512, −5.57584464736019374240099892814, −4.79626378000584497657631177613, −4.17581876295071181584436359786, −3.84219037711987742956936659638, −1.97975912387469931623681680626, −1.17512807566273782748131520780, 0,
1.17512807566273782748131520780, 1.97975912387469931623681680626, 3.84219037711987742956936659638, 4.17581876295071181584436359786, 4.79626378000584497657631177613, 5.57584464736019374240099892814, 6.19016337796064430599673156512, 6.61535538693502859985042344789, 7.77969292181990310865578301524