| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 7-s − 2·10-s + 3·11-s − 6·13-s − 2·14-s − 4·16-s − 19-s − 2·20-s + 6·22-s + 25-s − 12·26-s − 2·28-s − 5·29-s + 8·31-s − 8·32-s + 35-s − 2·37-s − 2·38-s + 5·41-s + 4·43-s + 6·44-s + 49-s + 2·50-s − 12·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s − 0.632·10-s + 0.904·11-s − 1.66·13-s − 0.534·14-s − 16-s − 0.229·19-s − 0.447·20-s + 1.27·22-s + 1/5·25-s − 2.35·26-s − 0.377·28-s − 0.928·29-s + 1.43·31-s − 1.41·32-s + 0.169·35-s − 0.328·37-s − 0.324·38-s + 0.780·41-s + 0.609·43-s + 0.904·44-s + 1/7·49-s + 0.282·50-s − 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.11405925737791, −13.68376050787083, −13.05042547110452, −12.63168565938630, −12.17609549374881, −11.93025078954172, −11.37820709159872, −10.88669355449627, −10.10698155759580, −9.693108040448747, −9.107184775898363, −8.709113275413110, −7.794893723507855, −7.432379373319231, −6.785012935076906, −6.410763237958968, −5.844824784293856, −5.154656760428185, −4.776264075956214, −4.138436437278321, −3.813945127540209, −3.094594417061698, −2.543185443658564, −2.010274420913059, −0.8710955061710361, 0,
0.8710955061710361, 2.010274420913059, 2.543185443658564, 3.094594417061698, 3.813945127540209, 4.138436437278321, 4.776264075956214, 5.154656760428185, 5.844824784293856, 6.410763237958968, 6.785012935076906, 7.432379373319231, 7.794893723507855, 8.709113275413110, 9.107184775898363, 9.693108040448747, 10.10698155759580, 10.88669355449627, 11.37820709159872, 11.93025078954172, 12.17609549374881, 12.63168565938630, 13.05042547110452, 13.68376050787083, 14.11405925737791
