| L(s) = 1 | − 2-s + 3-s + 4-s − 0.404·5-s − 6-s + 7-s − 8-s + 9-s + 0.404·10-s + 4.72·11-s + 12-s − 14-s − 0.404·15-s + 16-s + 4.50·17-s − 18-s − 4.94·19-s − 0.404·20-s + 21-s − 4.72·22-s − 5.73·23-s − 24-s − 4.83·25-s + 27-s + 28-s − 0.647·29-s + 0.404·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.180·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.127·10-s + 1.42·11-s + 0.288·12-s − 0.267·14-s − 0.104·15-s + 0.250·16-s + 1.09·17-s − 0.235·18-s − 1.13·19-s − 0.0903·20-s + 0.218·21-s − 1.00·22-s − 1.19·23-s − 0.204·24-s − 0.967·25-s + 0.192·27-s + 0.188·28-s − 0.120·29-s + 0.0737·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.404T + 5T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 + 0.647T + 29T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 + 4.41T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 + 0.666T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + 0.811T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 + 5.97T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 3.92T + 73T^{2} \) |
| 79 | \( 1 + 8.69T + 79T^{2} \) |
| 83 | \( 1 - 4.51T + 83T^{2} \) |
| 89 | \( 1 + 9.32T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−7.73157477392141410391107604035, −7.08453526197774797242179719665, −6.31862053431413430467973835621, −5.68961918655127229463325643877, −4.54151108525250126531973298475, −3.81077118894108223063023106299, −3.20571693566830043715708772945, −1.86851771403652240783641035349, −1.55434028370632662689865883698, 0,
1.55434028370632662689865883698, 1.86851771403652240783641035349, 3.20571693566830043715708772945, 3.81077118894108223063023106299, 4.54151108525250126531973298475, 5.68961918655127229463325643877, 6.31862053431413430467973835621, 7.08453526197774797242179719665, 7.73157477392141410391107604035
