| L(s) = 1 | + 0.342·2-s − 1.88·4-s + 0.548·5-s + 1.82·7-s − 1.32·8-s + 0.187·10-s − 5.99·11-s + 3.70·13-s + 0.624·14-s + 3.31·16-s + 1.64·17-s − 3.15·19-s − 1.03·20-s − 2.05·22-s + 5.46·23-s − 4.69·25-s + 1.26·26-s − 3.43·28-s − 7.24·29-s − 9.41·31-s + 3.79·32-s + 0.562·34-s + 1.00·35-s + 1.27·37-s − 1.08·38-s − 0.729·40-s + 5.81·41-s + ⋯ |
| L(s) = 1 | + 0.241·2-s − 0.941·4-s + 0.245·5-s + 0.689·7-s − 0.469·8-s + 0.0593·10-s − 1.80·11-s + 1.02·13-s + 0.166·14-s + 0.827·16-s + 0.398·17-s − 0.724·19-s − 0.231·20-s − 0.437·22-s + 1.13·23-s − 0.939·25-s + 0.248·26-s − 0.649·28-s − 1.34·29-s − 1.69·31-s + 0.669·32-s + 0.0964·34-s + 0.169·35-s + 0.209·37-s − 0.175·38-s − 0.115·40-s + 0.907·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.342T + 2T^{2} \) |
| 5 | \( 1 - 0.548T + 5T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 5.99T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 + 9.41T + 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 + 2.61T + 47T^{2} \) |
| 53 | \( 1 + 8.81T + 53T^{2} \) |
| 59 | \( 1 - 7.78T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 6.25T + 83T^{2} \) |
| 89 | \( 1 - 3.94T + 89T^{2} \) |
| 97 | \( 1 + 2.22T + 97T^{2} \) |
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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
−8.726655654645728571100829209376, −7.902555424853880361350168077217, −7.41690024004311747266430267727, −5.86324859350983323443449083348, −5.56393887557537570253691082806, −4.69643216471985829898091143750, −3.84959685910652872842972135807, −2.84217962103428799588343641783, −1.58078894028629667691367174478, 0,
1.58078894028629667691367174478, 2.84217962103428799588343641783, 3.84959685910652872842972135807, 4.69643216471985829898091143750, 5.56393887557537570253691082806, 5.86324859350983323443449083348, 7.41690024004311747266430267727, 7.902555424853880361350168077217, 8.726655654645728571100829209376
