| L(s) = 1 | + 2-s − 3.05·3-s + 4-s + 0.104·5-s − 3.05·6-s − 2.07·7-s + 8-s + 6.32·9-s + 0.104·10-s − 4.53·11-s − 3.05·12-s + 0.885·13-s − 2.07·14-s − 0.319·15-s + 16-s + 1.63·17-s + 6.32·18-s + 3.84·19-s + 0.104·20-s + 6.32·21-s − 4.53·22-s − 4.86·23-s − 3.05·24-s − 4.98·25-s + 0.885·26-s − 10.1·27-s − 2.07·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.76·3-s + 0.5·4-s + 0.0468·5-s − 1.24·6-s − 0.782·7-s + 0.353·8-s + 2.10·9-s + 0.0330·10-s − 1.36·11-s − 0.881·12-s + 0.245·13-s − 0.553·14-s − 0.0824·15-s + 0.250·16-s + 0.395·17-s + 1.49·18-s + 0.881·19-s + 0.0234·20-s + 1.37·21-s − 0.966·22-s − 1.01·23-s − 0.623·24-s − 0.997·25-s + 0.173·26-s − 1.95·27-s − 0.391·28-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 | \( 1 + 3.05T + 3T^{2} \) |
| 5 | \( 1 - 0.104T + 5T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 - 0.885T + 13T^{2} \) |
| 17 | \( 1 - 1.63T + 17T^{2} \) |
| 19 | \( 1 - 3.84T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 37 | \( 1 - 9.92T + 37T^{2} \) |
| 41 | \( 1 - 1.88T + 41T^{2} \) |
| 43 | \( 1 - 6.37T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 + 3.36T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 + 5.95T + 67T^{2} \) |
| 71 | \( 1 - 2.22T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 4.96T + 89T^{2} \) |
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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
−7.63934020929920412521373670039, −6.62185133396610878305515362407, −6.10083261542817389096309156940, −5.68404874583475448371845450000, −4.95829696267398815091779369262, −4.34013159029307782561123569096, −3.36470298392544809678410656341, −2.42571245307721357004082221536, −1.09733268910164830346154749861, 0,
1.09733268910164830346154749861, 2.42571245307721357004082221536, 3.36470298392544809678410656341, 4.34013159029307782561123569096, 4.95829696267398815091779369262, 5.68404874583475448371845450000, 6.10083261542817389096309156940, 6.62185133396610878305515362407, 7.63934020929920412521373670039
