| L(s) = 1 | + 2.21·2-s + 2.90·4-s − 5-s + 3.83·7-s + 2·8-s − 2.21·10-s + 5.80·11-s + 8.49·14-s − 1.37·16-s + 3.59·17-s − 2.14·19-s − 2.90·20-s + 12.8·22-s + 6.23·23-s + 25-s + 11.1·28-s − 2.06·29-s + 5.28·31-s − 7.05·32-s + 7.95·34-s − 3.83·35-s + 4.75·37-s − 4.75·38-s − 2·40-s − 11.1·41-s − 9.49·43-s + 16.8·44-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.45·4-s − 0.447·5-s + 1.45·7-s + 0.707·8-s − 0.700·10-s + 1.75·11-s + 2.27·14-s − 0.344·16-s + 0.871·17-s − 0.492·19-s − 0.649·20-s + 2.74·22-s + 1.30·23-s + 0.200·25-s + 2.10·28-s − 0.383·29-s + 0.948·31-s − 1.24·32-s + 1.36·34-s − 0.648·35-s + 0.781·37-s − 0.771·38-s − 0.316·40-s − 1.73·41-s − 1.44·43-s + 2.54·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.656951296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.656951296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 17 | \( 1 - 3.59T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 - 4.75T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 - 2.21T + 47T^{2} \) |
| 53 | \( 1 - 0.815T + 53T^{2} \) |
| 59 | \( 1 + 9.97T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 - 3.55T + 67T^{2} \) |
| 71 | \( 1 - 5.73T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 0.622T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 4.25T + 97T^{2} \) |
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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
−7.75498326219647882924291924501, −6.82084161548455088745730280525, −6.51714212869250073008800515788, −5.50304928504244658353235675237, −4.93136453135843298261428708654, −4.38023636974879115456870127758, −3.72807689523399244204625724895, −3.06025163313668731879141817195, −1.90773578449243501622159322919, −1.13230199209152852719195483170,
1.13230199209152852719195483170, 1.90773578449243501622159322919, 3.06025163313668731879141817195, 3.72807689523399244204625724895, 4.38023636974879115456870127758, 4.93136453135843298261428708654, 5.50304928504244658353235675237, 6.51714212869250073008800515788, 6.82084161548455088745730280525, 7.75498326219647882924291924501
