| L(s) = 1 | − 1.99·2-s − 3.03·3-s + 1.96·4-s − 1.96·5-s + 6.04·6-s + 0.618·7-s + 0.0731·8-s + 6.21·9-s + 3.91·10-s − 1.03·11-s − 5.95·12-s − 1.13·13-s − 1.23·14-s + 5.96·15-s − 4.07·16-s + 1.84·17-s − 12.3·18-s + 5.19·19-s − 3.85·20-s − 1.87·21-s + 2.06·22-s − 0.0424·23-s − 0.222·24-s − 1.13·25-s + 2.25·26-s − 9.74·27-s + 1.21·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 1.75·3-s + 0.981·4-s − 0.878·5-s + 2.46·6-s + 0.233·7-s + 0.0258·8-s + 2.07·9-s + 1.23·10-s − 0.312·11-s − 1.72·12-s − 0.313·13-s − 0.328·14-s + 1.53·15-s − 1.01·16-s + 0.447·17-s − 2.91·18-s + 1.19·19-s − 0.862·20-s − 0.409·21-s + 0.439·22-s − 0.00884·23-s − 0.0453·24-s − 0.227·25-s + 0.441·26-s − 1.87·27-s + 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 0.0424T + 23T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 + 2.74T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 + 0.286T + 43T^{2} \) |
| 47 | \( 1 + 0.412T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 5.80T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 + 1.88T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 0.989T + 97T^{2} \) |
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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
−7.917278486884935289232159396400, −7.37999276435240793576579960264, −6.95909692794889126096806304453, −5.89931765151816583165240332111, −5.18626170724414604567220109177, −4.51561901553490325690823616576, −3.47476037932414643026362563337, −1.84575903089476817692224124360, −0.847779110318817791616483162454, 0,
0.847779110318817791616483162454, 1.84575903089476817692224124360, 3.47476037932414643026362563337, 4.51561901553490325690823616576, 5.18626170724414604567220109177, 5.89931765151816583165240332111, 6.95909692794889126096806304453, 7.37999276435240793576579960264, 7.917278486884935289232159396400
