| L(s) = 1 | − 0.682·2-s − 2.48·3-s − 1.53·4-s + 1.69·6-s − 0.856·7-s + 2.41·8-s + 3.17·9-s + 11-s + 3.80·12-s − 5.50·13-s + 0.585·14-s + 1.41·16-s + 3.50·17-s − 2.16·18-s + 19-s + 2.12·21-s − 0.682·22-s − 6.75·23-s − 5.99·24-s + 3.76·26-s − 0.425·27-s + 1.31·28-s − 0.474·29-s − 5.19·31-s − 5.79·32-s − 2.48·33-s − 2.39·34-s + ⋯ |
| L(s) = 1 | − 0.482·2-s − 1.43·3-s − 0.766·4-s + 0.692·6-s − 0.323·7-s + 0.853·8-s + 1.05·9-s + 0.301·11-s + 1.09·12-s − 1.52·13-s + 0.156·14-s + 0.354·16-s + 0.850·17-s − 0.510·18-s + 0.229·19-s + 0.464·21-s − 0.145·22-s − 1.40·23-s − 1.22·24-s + 0.737·26-s − 0.0818·27-s + 0.248·28-s − 0.0881·29-s − 0.933·31-s − 1.02·32-s − 0.432·33-s − 0.410·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.682T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 7 | \( 1 + 0.856T + 7T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 + 0.474T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 + 0.420T + 37T^{2} \) |
| 41 | \( 1 - 8.25T + 41T^{2} \) |
| 43 | \( 1 - 1.53T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 - 4.79T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 3.08T + 89T^{2} \) |
| 97 | \( 1 - 8.88T + 97T^{2} \) |
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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
−7.67942984550721596878348135567, −7.26421808831494547862240668609, −6.28726142597116062448606062023, −5.61636005485671991860587112112, −5.03938240128567783466111555706, −4.35668272808053105497259883321, −3.48548855358306797990512561822, −2.08515028975708043599609357709, −0.853220985494044813489631171782, 0,
0.853220985494044813489631171782, 2.08515028975708043599609357709, 3.48548855358306797990512561822, 4.35668272808053105497259883321, 5.03938240128567783466111555706, 5.61636005485671991860587112112, 6.28726142597116062448606062023, 7.26421808831494547862240668609, 7.67942984550721596878348135567
