| L(s) = 1 | + 1.09·2-s − 3.36·3-s − 0.798·4-s − 3.68·6-s + 1.59·7-s − 3.06·8-s + 8.32·9-s − 3.68·11-s + 2.68·12-s + 1.74·14-s − 1.76·16-s − 2.98·17-s + 9.11·18-s + 2.79·19-s − 5.35·21-s − 4.04·22-s − 0.671·23-s + 10.3·24-s − 17.9·27-s − 1.27·28-s + 2.75·29-s + 6.43·31-s + 4.20·32-s + 12.4·33-s − 3.27·34-s − 6.64·36-s + 0.958·37-s + ⋯ |
| L(s) = 1 | + 0.774·2-s − 1.94·3-s − 0.399·4-s − 1.50·6-s + 0.601·7-s − 1.08·8-s + 2.77·9-s − 1.11·11-s + 0.775·12-s + 0.466·14-s − 0.441·16-s − 0.723·17-s + 2.14·18-s + 0.642·19-s − 1.16·21-s − 0.861·22-s − 0.139·23-s + 2.10·24-s − 3.44·27-s − 0.240·28-s + 0.512·29-s + 1.15·31-s + 0.742·32-s + 2.15·33-s − 0.561·34-s − 1.10·36-s + 0.157·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 3 | \( 1 + 3.36T + 3T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 - 2.79T + 19T^{2} \) |
| 23 | \( 1 + 0.671T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 0.958T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 - 4.88T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 8.57T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 1.65T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−7.87759176533845169454524880725, −6.98859915633773365334949482754, −6.29025245869563268430462321364, −5.59531748007868848349978376218, −5.03365993944984267774461053996, −4.63972659418231085523949490588, −3.83500461561312530081840889078, −2.49866816797070455616163761543, −1.06795821610517689851939309768, 0,
1.06795821610517689851939309768, 2.49866816797070455616163761543, 3.83500461561312530081840889078, 4.63972659418231085523949490588, 5.03365993944984267774461053996, 5.59531748007868848349978376218, 6.29025245869563268430462321364, 6.98859915633773365334949482754, 7.87759176533845169454524880725
