| L(s) = 1 | − 0.975·2-s + 1.70·3-s − 1.04·4-s + 5-s − 1.66·6-s + 0.984·7-s + 2.97·8-s − 0.0798·9-s − 0.975·10-s − 11-s − 1.79·12-s − 5.39·13-s − 0.960·14-s + 1.70·15-s − 0.803·16-s − 1.91·17-s + 0.0778·18-s + 4.82·19-s − 1.04·20-s + 1.68·21-s + 0.975·22-s + 6.73·23-s + 5.08·24-s + 25-s + 5.26·26-s − 5.26·27-s − 1.03·28-s + ⋯ |
| L(s) = 1 | − 0.689·2-s + 0.986·3-s − 0.524·4-s + 0.447·5-s − 0.680·6-s + 0.372·7-s + 1.05·8-s − 0.0266·9-s − 0.308·10-s − 0.301·11-s − 0.517·12-s − 1.49·13-s − 0.256·14-s + 0.441·15-s − 0.200·16-s − 0.465·17-s + 0.0183·18-s + 1.10·19-s − 0.234·20-s + 0.367·21-s + 0.207·22-s + 1.40·23-s + 1.03·24-s + 0.200·25-s + 1.03·26-s − 1.01·27-s − 0.195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 0.975T + 2T^{2} \) |
| 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 - 0.984T + 7T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 + 9.97T + 31T^{2} \) |
| 37 | \( 1 - 3.03T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 1.37T + 43T^{2} \) |
| 47 | \( 1 + 2.94T + 47T^{2} \) |
| 53 | \( 1 + 0.419T + 53T^{2} \) |
| 59 | \( 1 + 2.03T + 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 + 6.04T + 67T^{2} \) |
| 71 | \( 1 + 1.32T + 71T^{2} \) |
| 79 | \( 1 - 6.33T + 79T^{2} \) |
| 83 | \( 1 + 7.28T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.109387535634143138927745060101, −7.52093755515511967711434886641, −7.14191308469849739518392766840, −5.62676626461904678871920985317, −5.11784389270395779289286542929, −4.26247251126832226234867695378, −3.20076398991542368853415663921, −2.39532573157658004865102972835, −1.47957691987802574913676625468, 0,
1.47957691987802574913676625468, 2.39532573157658004865102972835, 3.20076398991542368853415663921, 4.26247251126832226234867695378, 5.11784389270395779289286542929, 5.62676626461904678871920985317, 7.14191308469849739518392766840, 7.52093755515511967711434886641, 8.109387535634143138927745060101
