| L(s) = 1 | − 0.802·2-s + 1.76·3-s − 1.35·4-s − 1.41·6-s − 0.592·7-s + 2.69·8-s + 0.100·9-s − 2.38·12-s + 1.79·13-s + 0.475·14-s + 0.549·16-s − 7.07·17-s − 0.0804·18-s + 2.28·19-s − 1.04·21-s + 1.49·23-s + 4.74·24-s − 1.44·26-s − 5.10·27-s + 0.802·28-s − 3.57·29-s + 6.16·31-s − 5.82·32-s + 5.68·34-s − 0.135·36-s + 7.33·37-s − 1.83·38-s + ⋯ |
| L(s) = 1 | − 0.567·2-s + 1.01·3-s − 0.677·4-s − 0.576·6-s − 0.223·7-s + 0.952·8-s + 0.0334·9-s − 0.689·12-s + 0.497·13-s + 0.127·14-s + 0.137·16-s − 1.71·17-s − 0.0189·18-s + 0.524·19-s − 0.227·21-s + 0.310·23-s + 0.968·24-s − 0.282·26-s − 0.982·27-s + 0.151·28-s − 0.663·29-s + 1.10·31-s − 1.03·32-s + 0.974·34-s − 0.0226·36-s + 1.20·37-s − 0.297·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.802T + 2T^{2} \) |
| 3 | \( 1 - 1.76T + 3T^{2} \) |
| 7 | \( 1 + 0.592T + 7T^{2} \) |
| 13 | \( 1 - 1.79T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 - 7.33T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 + 9.51T + 43T^{2} \) |
| 47 | \( 1 - 1.93T + 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 59 | \( 1 + 0.0382T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 - 6.79T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 4.52T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 6.21T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−8.594912120393656270831500875708, −7.906181437840370582934179067851, −7.07281975511865423050864874380, −6.20776474005894501169473474351, −5.11686813230088232658016697301, −4.29944598384956607194848455244, −3.49816581009018571047675296673, −2.57802977422287328239640093445, −1.48317638671443727267099649017, 0,
1.48317638671443727267099649017, 2.57802977422287328239640093445, 3.49816581009018571047675296673, 4.29944598384956607194848455244, 5.11686813230088232658016697301, 6.20776474005894501169473474351, 7.07281975511865423050864874380, 7.906181437840370582934179067851, 8.594912120393656270831500875708
