| L(s) = 1 | + 2·3-s − 5-s + 7-s + 9-s + 11-s − 4·13-s − 2·15-s + 17-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 + 2·51-s − 7·53-s − 55-s − 7·59-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 + 0.242·17-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 + 0.280·51-s − 0.961·53-s − 0.134·55-s − 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 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
−14.54247408682646, −14.15734542100849, −13.86736156608019, −12.93053915634877, −12.65209233994844, −12.12486997292216, −11.52177107984960, −11.05951934476727, −10.45097795285460, −9.800097498913885, −9.332041273792462, −8.930584328430085, −8.313912337597581, −7.803826099309470, −7.515526030305819, −6.888338092998862, −6.215855798611687, −5.403736208042329, −4.936557098408172, −4.222663197192505, −3.629816378087200, −3.193545865738773, −2.299573514699404, −2.061372212987541, −1.007894419511678, 0,
1.007894419511678, 2.061372212987541, 2.299573514699404, 3.193545865738773, 3.629816378087200, 4.222663197192505, 4.936557098408172, 5.403736208042329, 6.215855798611687, 6.888338092998862, 7.515526030305819, 7.803826099309470, 8.313912337597581, 8.930584328430085, 9.332041273792462, 9.800097498913885, 10.45097795285460, 11.05951934476727, 11.52177107984960, 12.12486997292216, 12.65209233994844, 12.93053915634877, 13.86736156608019, 14.15734542100849, 14.54247408682646
