| L(s) = 1 | − 1.62·2-s + 0.656·4-s + 5-s − 1.43·7-s + 2.18·8-s − 1.62·10-s − 6.20·11-s + 3.99·13-s + 2.33·14-s − 4.88·16-s + 5.83·17-s − 8.48·19-s + 0.656·20-s + 10.1·22-s − 1.19·23-s + 25-s − 6.50·26-s − 0.941·28-s + 4.83·29-s + 9.78·31-s + 3.57·32-s − 9.51·34-s − 1.43·35-s + 0.211·37-s + 13.8·38-s + 2.18·40-s + 5.06·41-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.328·4-s + 0.447·5-s − 0.542·7-s + 0.774·8-s − 0.515·10-s − 1.87·11-s + 1.10·13-s + 0.624·14-s − 1.22·16-s + 1.41·17-s − 1.94·19-s + 0.146·20-s + 2.15·22-s − 0.248·23-s + 0.200·25-s − 1.27·26-s − 0.177·28-s + 0.896·29-s + 1.75·31-s + 0.632·32-s − 1.63·34-s − 0.242·35-s + 0.0347·37-s + 2.24·38-s + 0.346·40-s + 0.790·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 - 9.78T + 31T^{2} \) |
| 37 | \( 1 - 0.211T + 37T^{2} \) |
| 41 | \( 1 - 5.06T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 + 3.37T + 47T^{2} \) |
| 53 | \( 1 - 7.97T + 53T^{2} \) |
| 59 | \( 1 + 8.77T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 9.96T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 - 2.27T + 83T^{2} \) |
| 97 | \( 1 - 0.714T + 97T^{2} \) |
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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
−8.197732723495482991476805223790, −7.75543166064777217118702157016, −6.61964365485440411860229978172, −6.09046811958223709925938144926, −5.13569233678608261637045484076, −4.33727564214055265602273050774, −3.13963572724921234409231828328, −2.30360601643444887262777686300, −1.16487096281072099295939836800, 0,
1.16487096281072099295939836800, 2.30360601643444887262777686300, 3.13963572724921234409231828328, 4.33727564214055265602273050774, 5.13569233678608261637045484076, 6.09046811958223709925938144926, 6.61964365485440411860229978172, 7.75543166064777217118702157016, 8.197732723495482991476805223790
