| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 2·7-s − 8-s + 9-s − 2·10-s − 4·11-s − 12-s − 2·13-s + 2·14-s − 2·15-s + 16-s − 2·17-s − 18-s + 2·20-s + 2·21-s + 4·22-s + 2·23-s + 24-s − 25-s + 2·26-s − 27-s − 2·28-s − 10·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.447·20-s + 0.436·21-s + 0.852·22-s + 0.417·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.90158891974630, −13.41686148664715, −13.07427283338908, −12.71205544617169, −12.02598124422563, −11.58068906439512, −10.87203343581195, −10.64060837961075, −10.00092930139284, −9.702425785534607, −9.257808511778566, −8.754646341277421, −7.823326887924585, −7.689107814242713, −6.914460141850708, −6.490072776788374, −5.919382787581470, −5.486053298561149, −4.993893351433892, −4.248674118953603, −3.444379749178849, −2.719530708449666, −2.260916966641820, −1.648598660615669, −0.6591652694219996, 0,
0.6591652694219996, 1.648598660615669, 2.260916966641820, 2.719530708449666, 3.444379749178849, 4.248674118953603, 4.993893351433892, 5.486053298561149, 5.919382787581470, 6.490072776788374, 6.914460141850708, 7.689107814242713, 7.823326887924585, 8.754646341277421, 9.257808511778566, 9.702425785534607, 10.00092930139284, 10.64060837961075, 10.87203343581195, 11.58068906439512, 12.02598124422563, 12.71205544617169, 13.07427283338908, 13.41686148664715, 13.90158891974630
