| L(s) = 1 | − 1.94·2-s + 1.79·4-s + 5-s − 3.59·7-s + 0.405·8-s − 1.94·10-s + 4.44·11-s + 7.00·14-s − 4.37·16-s + 0.663·17-s − 7.05·19-s + 1.79·20-s − 8.65·22-s − 8.85·23-s + 25-s − 6.44·28-s − 0.195·29-s + 5.38·31-s + 7.70·32-s − 1.29·34-s − 3.59·35-s + 9.51·37-s + 13.7·38-s + 0.405·40-s + 10.8·41-s + 8.75·43-s + 7.96·44-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.895·4-s + 0.447·5-s − 1.35·7-s + 0.143·8-s − 0.615·10-s + 1.34·11-s + 1.87·14-s − 1.09·16-s + 0.160·17-s − 1.61·19-s + 0.400·20-s − 1.84·22-s − 1.84·23-s + 0.200·25-s − 1.21·28-s − 0.0362·29-s + 0.966·31-s + 1.36·32-s − 0.221·34-s − 0.608·35-s + 1.56·37-s + 2.22·38-s + 0.0640·40-s + 1.70·41-s + 1.33·43-s + 1.20·44-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 17 | \( 1 - 0.663T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 + 0.195T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.75T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 0.285T + 71T^{2} \) |
| 73 | \( 1 - 6.16T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−7.76647304153185435248372580401, −6.81311246113455624854232907338, −6.18013643361697658422720053726, −6.07515547008972844491197193296, −4.37376309042269807443016199638, −4.05571253036572508726377322185, −2.80442114867487320906889162009, −2.03749793659674753977082855877, −1.04198720411727527596640439106, 0,
1.04198720411727527596640439106, 2.03749793659674753977082855877, 2.80442114867487320906889162009, 4.05571253036572508726377322185, 4.37376309042269807443016199638, 6.07515547008972844491197193296, 6.18013643361697658422720053726, 6.81311246113455624854232907338, 7.76647304153185435248372580401
