| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 3·9-s − 2·10-s + 6·11-s + 4·13-s + 16-s + 6·17-s + 3·18-s − 8·19-s + 2·20-s − 6·22-s − 4·23-s − 25-s − 4·26-s − 2·29-s − 8·31-s − 32-s − 6·34-s − 3·36-s + 10·37-s + 8·38-s − 2·40-s − 2·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 9-s − 0.632·10-s + 1.80·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 1.83·19-s + 0.447·20-s − 1.27·22-s − 0.834·23-s − 1/5·25-s − 0.784·26-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.64·37-s + 1.29·38-s − 0.316·40-s − 0.312·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9425722449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9425722449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.13307129139539617936872976414, −11.87036840604066530631171049169, −11.06145732082171547076231887501, −9.875254168076625232465392939952, −9.023272714070285939939936952762, −8.153392726763725992888200891001, −6.43384872055430414742348619213, −5.87142146066002283835169948551, −3.67656046849713124200251791765, −1.73494139896824873361280163139,
1.73494139896824873361280163139, 3.67656046849713124200251791765, 5.87142146066002283835169948551, 6.43384872055430414742348619213, 8.153392726763725992888200891001, 9.023272714070285939939936952762, 9.875254168076625232465392939952, 11.06145732082171547076231887501, 11.87036840604066530631171049169, 13.13307129139539617936872976414
