| L(s) = 1 | + 2-s + 4-s − 4.06·5-s + 1.57·7-s + 8-s − 4.06·10-s − 5.58·11-s − 4.93·13-s + 1.57·14-s + 16-s + 7.87·17-s + 7.88·19-s − 4.06·20-s − 5.58·22-s − 0.940·23-s + 11.5·25-s − 4.93·26-s + 1.57·28-s − 0.204·29-s − 0.902·31-s + 32-s + 7.87·34-s − 6.39·35-s + 0.171·37-s + 7.88·38-s − 4.06·40-s − 4.80·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.81·5-s + 0.594·7-s + 0.353·8-s − 1.28·10-s − 1.68·11-s − 1.36·13-s + 0.420·14-s + 0.250·16-s + 1.90·17-s + 1.80·19-s − 0.909·20-s − 1.19·22-s − 0.196·23-s + 2.30·25-s − 0.968·26-s + 0.297·28-s − 0.0380·29-s − 0.162·31-s + 0.176·32-s + 1.35·34-s − 1.08·35-s + 0.0281·37-s + 1.27·38-s − 0.643·40-s − 0.750·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 4.06T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 - 7.87T + 17T^{2} \) |
| 19 | \( 1 - 7.88T + 19T^{2} \) |
| 23 | \( 1 + 0.940T + 23T^{2} \) |
| 29 | \( 1 + 0.204T + 29T^{2} \) |
| 31 | \( 1 + 0.902T + 31T^{2} \) |
| 37 | \( 1 - 0.171T + 37T^{2} \) |
| 41 | \( 1 + 4.80T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 5.30T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 - 5.47T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 0.418T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 0.739T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 0.822T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−7.54775721372996326021431019432, −7.27102871056782302712451321405, −5.78999926805846972128188636381, −5.12571027167618745720479444892, −4.83523690813500699078773487856, −3.89856342792538351611727267306, −3.11424132196256505029629786525, −2.69401571871987005747759398632, −1.18935626398359968054675704838, 0,
1.18935626398359968054675704838, 2.69401571871987005747759398632, 3.11424132196256505029629786525, 3.89856342792538351611727267306, 4.83523690813500699078773487856, 5.12571027167618745720479444892, 5.78999926805846972128188636381, 7.27102871056782302712451321405, 7.54775721372996326021431019432
