L(s) = 1 | + 0.338·2-s + 0.415·3-s − 1.88·4-s + 2.48·5-s + 0.140·6-s + 1.37·7-s − 1.31·8-s − 2.82·9-s + 0.842·10-s + 11-s − 0.782·12-s − 6.16·13-s + 0.464·14-s + 1.03·15-s + 3.32·16-s − 6.22·17-s − 0.958·18-s − 1.56·19-s − 4.68·20-s + 0.569·21-s + 0.338·22-s − 8.96·23-s − 0.546·24-s + 1.18·25-s − 2.09·26-s − 2.41·27-s − 2.58·28-s + ⋯ |
L(s) = 1 | + 0.239·2-s + 0.239·3-s − 0.942·4-s + 1.11·5-s + 0.0574·6-s + 0.518·7-s − 0.465·8-s − 0.942·9-s + 0.266·10-s + 0.301·11-s − 0.225·12-s − 1.71·13-s + 0.124·14-s + 0.266·15-s + 0.830·16-s − 1.50·17-s − 0.225·18-s − 0.359·19-s − 1.04·20-s + 0.124·21-s + 0.0722·22-s − 1.87·23-s − 0.111·24-s + 0.236·25-s − 0.410·26-s − 0.465·27-s − 0.488·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 - 0.338T + 2T^{2} \) |
| 3 | \( 1 - 0.415T + 3T^{2} \) |
| 5 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 13 | \( 1 + 6.16T + 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 + 8.96T + 23T^{2} \) |
| 29 | \( 1 - 0.948T + 29T^{2} \) |
| 31 | \( 1 - 9.67T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 + 9.20T + 41T^{2} \) |
| 43 | \( 1 - 1.99T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 0.208T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 0.383T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 + 5.72T + 89T^{2} \) |
| 97 | \( 1 - 7.59T + 97T^{2} \) |
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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
−9.061811684785377215962391006203, −8.540145264607385820885226966358, −7.68838829936442250962379923547, −6.40684021302317291689859230052, −5.77117588712804088861588234753, −4.83820920366135048684888351016, −4.24639150197373705617632048545, −2.75958526171787397157693518539, −1.99129334535582377614296581428, 0,
1.99129334535582377614296581428, 2.75958526171787397157693518539, 4.24639150197373705617632048545, 4.83820920366135048684888351016, 5.77117588712804088861588234753, 6.40684021302317291689859230052, 7.68838829936442250962379923547, 8.540145264607385820885226966358, 9.061811684785377215962391006203