| L(s) = 1 | + 2.37·2-s + 0.134·3-s + 3.61·4-s + 0.317·6-s + 7-s + 3.83·8-s − 2.98·9-s − 4.97·11-s + 0.484·12-s − 4.25·13-s + 2.37·14-s + 1.85·16-s − 1.45·17-s − 7.06·18-s − 2.16·19-s + 0.134·21-s − 11.7·22-s − 23-s + 0.514·24-s − 10.0·26-s − 0.801·27-s + 3.61·28-s + 3.40·29-s − 5.59·31-s − 3.27·32-s − 0.666·33-s − 3.45·34-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.0773·3-s + 1.80·4-s + 0.129·6-s + 0.377·7-s + 1.35·8-s − 0.994·9-s − 1.49·11-s + 0.139·12-s − 1.17·13-s + 0.633·14-s + 0.463·16-s − 0.353·17-s − 1.66·18-s − 0.496·19-s + 0.0292·21-s − 2.51·22-s − 0.208·23-s + 0.104·24-s − 1.97·26-s − 0.154·27-s + 0.683·28-s + 0.631·29-s − 1.00·31-s − 0.578·32-s − 0.116·33-s − 0.592·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 3 | \( 1 - 0.134T + 3T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 29 | \( 1 - 3.40T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 - 0.861T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 2.71T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 4.32T + 53T^{2} \) |
| 59 | \( 1 - 9.64T + 59T^{2} \) |
| 61 | \( 1 - 1.18T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 + 0.370T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.87801678866202708161869579531, −7.21051304354262514462134485287, −6.31649904645405429549949317021, −5.58260266456068212526556028438, −5.03731099416314427258791564284, −4.48963009961345388596156148236, −3.42687725036980584703576064353, −2.60474311415624372863211381662, −2.16918232577110310792588586099, 0,
2.16918232577110310792588586099, 2.60474311415624372863211381662, 3.42687725036980584703576064353, 4.48963009961345388596156148236, 5.03731099416314427258791564284, 5.58260266456068212526556028438, 6.31649904645405429549949317021, 7.21051304354262514462134485287, 7.87801678866202708161869579531
