| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 2·13-s − 14-s + 16-s − 4·19-s + 8·23-s − 5·25-s − 2·26-s + 28-s + 8·29-s − 8·31-s − 32-s + 8·37-s + 4·38-s + 8·41-s + 4·43-s − 8·46-s + 8·47-s + 49-s + 5·50-s + 2·52-s − 10·53-s − 56-s − 8·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s + 1.66·23-s − 25-s − 0.392·26-s + 0.188·28-s + 1.48·29-s − 1.43·31-s − 0.176·32-s + 1.31·37-s + 0.648·38-s + 1.24·41-s + 0.609·43-s − 1.17·46-s + 1.16·47-s + 1/7·49-s + 0.707·50-s + 0.277·52-s − 1.37·53-s − 0.133·56-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.22804773081020, −14.65990910408162, −14.29078812034902, −13.53797091568270, −13.03335404180752, −12.49206681560278, −11.98660401508289, −11.21021059883916, −10.86742985885174, −10.62675852433987, −9.671949533581188, −9.309786719365258, −8.715877141966961, −8.299020617623343, −7.530665477636827, −7.280421793622296, −6.377264606180438, −6.006366898513139, −5.333238367494198, −4.434249081996783, −4.083780283552313, −3.001371458045609, −2.594882915547116, −1.608119975296114, −1.060890753615265, 0,
1.060890753615265, 1.608119975296114, 2.594882915547116, 3.001371458045609, 4.083780283552313, 4.434249081996783, 5.333238367494198, 6.006366898513139, 6.377264606180438, 7.280421793622296, 7.530665477636827, 8.299020617623343, 8.715877141966961, 9.309786719365258, 9.671949533581188, 10.62675852433987, 10.86742985885174, 11.21021059883916, 11.98660401508289, 12.49206681560278, 13.03335404180752, 13.53797091568270, 14.29078812034902, 14.65990910408162, 15.22804773081020
