| L(s) = 1 | + 0.0776·2-s − 3-s − 1.99·4-s − 2.20·5-s − 0.0776·6-s − 0.310·8-s + 9-s − 0.171·10-s + 2.91·11-s + 1.99·12-s − 13-s + 2.20·15-s + 3.96·16-s − 0.393·17-s + 0.0776·18-s + 0.124·19-s + 4.39·20-s + 0.226·22-s + 3.38·23-s + 0.310·24-s − 0.149·25-s − 0.0776·26-s − 27-s − 0.642·29-s + 0.171·30-s + 5.95·31-s + 0.928·32-s + ⋯ |
| L(s) = 1 | + 0.0549·2-s − 0.577·3-s − 0.996·4-s − 0.984·5-s − 0.0317·6-s − 0.109·8-s + 0.333·9-s − 0.0540·10-s + 0.879·11-s + 0.575·12-s − 0.277·13-s + 0.568·15-s + 0.990·16-s − 0.0955·17-s + 0.0183·18-s + 0.0286·19-s + 0.981·20-s + 0.0482·22-s + 0.705·23-s + 0.0633·24-s − 0.0298·25-s − 0.0152·26-s − 0.192·27-s − 0.119·29-s + 0.0312·30-s + 1.07·31-s + 0.164·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0776T + 2T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 17 | \( 1 + 0.393T + 17T^{2} \) |
| 19 | \( 1 - 0.124T + 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 + 0.642T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 - 4.18T + 37T^{2} \) |
| 41 | \( 1 - 0.434T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 + 2.39T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 7.55T + 73T^{2} \) |
| 79 | \( 1 + 2.37T + 79T^{2} \) |
| 83 | \( 1 + 9.78T + 83T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.771973036730289081310301028030, −8.110295277502926306676349547418, −7.26832727708432563850057191431, −6.42909418611153448757486265809, −5.46602948415415040348933671955, −4.56202895279708484596392590755, −4.07356074463668789305165711392, −3.09125135051013456642779194757, −1.23478285211611589466438529491, 0,
1.23478285211611589466438529491, 3.09125135051013456642779194757, 4.07356074463668789305165711392, 4.56202895279708484596392590755, 5.46602948415415040348933671955, 6.42909418611153448757486265809, 7.26832727708432563850057191431, 8.110295277502926306676349547418, 8.771973036730289081310301028030
