| L(s) = 1 | − 2·3-s − 5-s + 7-s + 9-s − 11-s + 4·13-s + 2·15-s − 2·21-s − 2·23-s + 25-s + 4·27-s + 5·29-s − 10·31-s + 2·33-s − 35-s − 4·37-s − 8·39-s − 3·41-s + 8·43-s − 45-s − 3·47-s − 6·49-s + 7·53-s + 55-s − 7·59-s + 14·61-s + 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s − 0.436·21-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 0.928·29-s − 1.79·31-s + 0.348·33-s − 0.169·35-s − 0.657·37-s − 1.28·39-s − 0.468·41-s + 1.21·43-s − 0.149·45-s − 0.437·47-s − 6/7·49-s + 0.961·53-s + 0.134·55-s − 0.911·59-s + 1.79·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.02627113583323, −12.39026153574698, −12.16497633591222, −11.60151559537762, −11.19463293268594, −10.85022221427812, −10.57214727132639, −9.966570967028039, −9.363736273766671, −8.802219649408713, −8.312093282990134, −8.024142730497428, −7.324241865295813, −6.806641126307828, −6.459443490305628, −5.781830135296777, −5.499636228768926, −4.982412954424017, −4.490725660370153, −3.766688974710085, −3.517039463986000, −2.665864622006659, −1.992277978234496, −1.283967342111441, −0.6922784974056535, 0,
0.6922784974056535, 1.283967342111441, 1.992277978234496, 2.665864622006659, 3.517039463986000, 3.766688974710085, 4.490725660370153, 4.982412954424017, 5.499636228768926, 5.781830135296777, 6.459443490305628, 6.806641126307828, 7.324241865295813, 8.024142730497428, 8.312093282990134, 8.802219649408713, 9.363736273766671, 9.966570967028039, 10.57214727132639, 10.85022221427812, 11.19463293268594, 11.60151559537762, 12.16497633591222, 12.39026153574698, 13.02627113583323
