| L(s) = 1 | + 0.337·2-s − 1.88·4-s − 0.914·5-s + 2.15·7-s − 1.31·8-s − 0.308·10-s − 11-s − 2.61·13-s + 0.727·14-s + 3.33·16-s − 3.47·17-s − 2.56·19-s + 1.72·20-s − 0.337·22-s + 8.97·23-s − 4.16·25-s − 0.880·26-s − 4.06·28-s + 3.11·29-s + 4.44·31-s + 3.74·32-s − 1.17·34-s − 1.97·35-s + 3.13·37-s − 0.866·38-s + 1.19·40-s − 0.799·41-s + ⋯ |
| L(s) = 1 | + 0.238·2-s − 0.943·4-s − 0.409·5-s + 0.815·7-s − 0.463·8-s − 0.0975·10-s − 0.301·11-s − 0.724·13-s + 0.194·14-s + 0.832·16-s − 0.842·17-s − 0.589·19-s + 0.385·20-s − 0.0718·22-s + 1.87·23-s − 0.832·25-s − 0.172·26-s − 0.768·28-s + 0.578·29-s + 0.798·31-s + 0.661·32-s − 0.200·34-s − 0.333·35-s + 0.515·37-s − 0.140·38-s + 0.189·40-s − 0.124·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.337T + 2T^{2} \) |
| 5 | \( 1 + 0.914T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 8.97T + 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 + 0.799T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 0.636T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 8.21T + 73T^{2} \) |
| 79 | \( 1 + 6.06T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 9.45T + 97T^{2} \) |
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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
−7.66684883613003040865606771409, −6.82027963849930320688090649359, −6.01259324466939554367218585870, −5.09336046057095873598358146957, −4.67671098387492880323241616091, −4.17553342474581038121461394191, −3.15099100737015281716123818899, −2.35757688298813904641479755219, −1.11381104558481919138723249528, 0,
1.11381104558481919138723249528, 2.35757688298813904641479755219, 3.15099100737015281716123818899, 4.17553342474581038121461394191, 4.67671098387492880323241616091, 5.09336046057095873598358146957, 6.01259324466939554367218585870, 6.82027963849930320688090649359, 7.66684883613003040865606771409
