| L(s) = 1 | − 1.23·3-s − 5-s − 1.47·9-s + 5.23·11-s − 13-s + 1.23·15-s + 2·17-s + 5.23·19-s + 1.23·23-s + 25-s + 5.52·27-s − 0.472·29-s + 7.70·31-s − 6.47·33-s + 0.472·37-s + 1.23·39-s + 6.94·41-s + 1.23·43-s + 1.47·45-s − 4.94·47-s − 7·49-s − 2.47·51-s + 6.94·53-s − 5.23·55-s − 6.47·57-s + 7.70·59-s + 4.47·61-s + ⋯ |
| L(s) = 1 | − 0.713·3-s − 0.447·5-s − 0.490·9-s + 1.57·11-s − 0.277·13-s + 0.319·15-s + 0.485·17-s + 1.20·19-s + 0.257·23-s + 0.200·25-s + 1.06·27-s − 0.0876·29-s + 1.38·31-s − 1.12·33-s + 0.0776·37-s + 0.197·39-s + 1.08·41-s + 0.188·43-s + 0.219·45-s − 0.721·47-s − 49-s − 0.346·51-s + 0.953·53-s − 0.706·55-s − 0.857·57-s + 1.00·59-s + 0.572·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079507700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079507700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 6.94T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−11.12951574925593938543374835261, −9.999785734386787146458714838782, −9.162217464044415556471179323061, −8.192157076782164240718754090039, −7.10972410829731160032125077882, −6.26429252151906941391582673532, −5.30818460703614388109561460911, −4.20943309108429795598214803603, −3.02971160706151924177536074890, −1.02731907916227138323289547118,
1.02731907916227138323289547118, 3.02971160706151924177536074890, 4.20943309108429795598214803603, 5.30818460703614388109561460911, 6.26429252151906941391582673532, 7.10972410829731160032125077882, 8.192157076782164240718754090039, 9.162217464044415556471179323061, 9.999785734386787146458714838782, 11.12951574925593938543374835261
