| L(s) = 1 | + 2.24·3-s − 5-s + 7-s + 2.02·9-s − 5.09·11-s − 13-s − 2.24·15-s − 2.63·17-s + 6.11·19-s + 2.24·21-s − 5.44·23-s + 25-s − 2.17·27-s − 6.47·29-s − 3.50·31-s − 11.4·33-s − 35-s + 0.458·37-s − 2.24·39-s − 7.15·41-s + 2.62·43-s − 2.02·45-s − 8.09·47-s + 49-s − 5.91·51-s + 3.74·53-s + 5.09·55-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 0.447·5-s + 0.377·7-s + 0.676·9-s − 1.53·11-s − 0.277·13-s − 0.578·15-s − 0.639·17-s + 1.40·19-s + 0.489·21-s − 1.13·23-s + 0.200·25-s − 0.419·27-s − 1.20·29-s − 0.629·31-s − 1.98·33-s − 0.169·35-s + 0.0753·37-s − 0.359·39-s − 1.11·41-s + 0.399·43-s − 0.302·45-s − 1.18·47-s + 0.142·49-s − 0.828·51-s + 0.515·53-s + 0.686·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 + 6.47T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 - 0.458T + 37T^{2} \) |
| 41 | \( 1 + 7.15T + 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 + 8.09T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 - 9.17T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 + 8.27T + 73T^{2} \) |
| 79 | \( 1 - 3.86T + 79T^{2} \) |
| 83 | \( 1 + 8.61T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 8.15T + 97T^{2} \) |
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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
−8.270048394468544072467375550938, −7.49707831284595023010415911747, −7.19413229125968558718428953911, −5.73874195854876828266030128154, −5.12793516326234278790497335556, −4.12591475985970370471108407296, −3.34094770951071722910562684366, −2.58325519144815968624140872546, −1.79061753565348062261397577800, 0,
1.79061753565348062261397577800, 2.58325519144815968624140872546, 3.34094770951071722910562684366, 4.12591475985970370471108407296, 5.12793516326234278790497335556, 5.73874195854876828266030128154, 7.19413229125968558718428953911, 7.49707831284595023010415911747, 8.270048394468544072467375550938
