L(s) = 1 | + 1.43·2-s + 0.0711·4-s − 2.31·5-s + 4.44·7-s − 2.77·8-s − 3.33·10-s − 4.52·11-s + 1.17·13-s + 6.39·14-s − 4.13·16-s + 4.86·19-s − 0.164·20-s − 6.51·22-s − 0.625·23-s + 0.354·25-s + 1.69·26-s + 0.316·28-s − 1.51·29-s − 8.73·31-s − 0.402·32-s − 10.2·35-s − 8.79·37-s + 7.00·38-s + 6.42·40-s − 0.464·41-s − 1.51·43-s − 0.321·44-s + ⋯ |
L(s) = 1 | + 1.01·2-s + 0.0355·4-s − 1.03·5-s + 1.67·7-s − 0.981·8-s − 1.05·10-s − 1.36·11-s + 0.325·13-s + 1.70·14-s − 1.03·16-s + 1.11·19-s − 0.0368·20-s − 1.38·22-s − 0.130·23-s + 0.0708·25-s + 0.331·26-s + 0.0597·28-s − 0.280·29-s − 1.56·31-s − 0.0711·32-s − 1.73·35-s − 1.44·37-s + 1.13·38-s + 1.01·40-s − 0.0726·41-s − 0.230·43-s − 0.0485·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 19 | \( 1 - 4.86T + 19T^{2} \) |
| 23 | \( 1 + 0.625T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 + 0.464T + 41T^{2} \) |
| 43 | \( 1 + 1.51T + 43T^{2} \) |
| 47 | \( 1 + 6.01T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 + 1.54T + 89T^{2} \) |
| 97 | \( 1 - 3.83T + 97T^{2} \) |
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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
−8.329909440870071619393310299029, −7.73058024932352073324204305013, −7.14438105437553299074281542387, −5.68557989674245591880923341934, −5.23431876116795500506905933343, −4.60308636331657089248095392796, −3.79088041201331679649847508804, −3.00252751878126545453657420363, −1.71094289357146942503382595001, 0,
1.71094289357146942503382595001, 3.00252751878126545453657420363, 3.79088041201331679649847508804, 4.60308636331657089248095392796, 5.23431876116795500506905933343, 5.68557989674245591880923341934, 7.14438105437553299074281542387, 7.73058024932352073324204305013, 8.329909440870071619393310299029