| L(s) = 1 | − 2-s + 1.29·3-s + 4-s + 3.44·5-s − 1.29·6-s − 0.471·7-s − 8-s − 1.33·9-s − 3.44·10-s − 1.26·11-s + 1.29·12-s + 2.78·13-s + 0.471·14-s + 4.43·15-s + 16-s − 6.13·17-s + 1.33·18-s + 0.199·19-s + 3.44·20-s − 0.608·21-s + 1.26·22-s − 8.85·23-s − 1.29·24-s + 6.84·25-s − 2.78·26-s − 5.59·27-s − 0.471·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.744·3-s + 0.5·4-s + 1.53·5-s − 0.526·6-s − 0.178·7-s − 0.353·8-s − 0.445·9-s − 1.08·10-s − 0.381·11-s + 0.372·12-s + 0.772·13-s + 0.126·14-s + 1.14·15-s + 0.250·16-s − 1.48·17-s + 0.314·18-s + 0.0457·19-s + 0.769·20-s − 0.132·21-s + 0.269·22-s − 1.84·23-s − 0.263·24-s + 1.36·25-s − 0.546·26-s − 1.07·27-s − 0.0891·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.29T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 + 0.471T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 + 6.13T + 17T^{2} \) |
| 19 | \( 1 - 0.199T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 1.29T + 47T^{2} \) |
| 53 | \( 1 - 7.50T + 53T^{2} \) |
| 59 | \( 1 - 3.31T + 59T^{2} \) |
| 61 | \( 1 - 0.0255T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 6.45T + 73T^{2} \) |
| 79 | \( 1 + 9.96T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
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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
−8.216780548060416192967976731808, −6.80602626099475320083523074883, −6.52838509105294863356000900465, −5.73028861377372198069706739499, −5.03686162321236880086342481618, −3.79003604406552923535436711208, −2.92679691503228252736319552047, −2.11947934285965506916329651129, −1.67834008673186090046496168084, 0,
1.67834008673186090046496168084, 2.11947934285965506916329651129, 2.92679691503228252736319552047, 3.79003604406552923535436711208, 5.03686162321236880086342481618, 5.73028861377372198069706739499, 6.52838509105294863356000900465, 6.80602626099475320083523074883, 8.216780548060416192967976731808
