| L(s) = 1 | + 2.11·2-s + 2.49·4-s + 5-s + 1.69·7-s + 1.03·8-s + 2.11·10-s + 1.48·11-s − 0.319·13-s + 3.59·14-s − 2.77·16-s + 2.82·17-s + 8.40·19-s + 2.49·20-s + 3.13·22-s − 5.00·23-s + 25-s − 0.676·26-s + 4.22·28-s + 6.70·29-s + 1.54·31-s − 7.96·32-s + 5.99·34-s + 1.69·35-s − 2.62·37-s + 17.8·38-s + 1.03·40-s − 0.927·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.24·4-s + 0.447·5-s + 0.640·7-s + 0.367·8-s + 0.670·10-s + 0.446·11-s − 0.0885·13-s + 0.959·14-s − 0.694·16-s + 0.686·17-s + 1.92·19-s + 0.556·20-s + 0.669·22-s − 1.04·23-s + 0.200·25-s − 0.132·26-s + 0.797·28-s + 1.24·29-s + 0.278·31-s − 1.40·32-s + 1.02·34-s + 0.286·35-s − 0.431·37-s + 2.88·38-s + 0.164·40-s − 0.144·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.559248348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.559248348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 + 0.319T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 8.40T + 19T^{2} \) |
| 23 | \( 1 + 5.00T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 1.54T + 31T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 + 0.927T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 - 7.52T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.232982112858895263400869753342, −7.60709569883525187957422711619, −6.62993899729066981399209566550, −6.10407999741757852839015798570, −5.13542684411188694930633259309, −4.96970053805699150091131278157, −3.84491943125142024340519908993, −3.21668192810324624995000154249, −2.26233345956329215087648707553, −1.18517020982926405293200383430,
1.18517020982926405293200383430, 2.26233345956329215087648707553, 3.21668192810324624995000154249, 3.84491943125142024340519908993, 4.96970053805699150091131278157, 5.13542684411188694930633259309, 6.10407999741757852839015798570, 6.62993899729066981399209566550, 7.60709569883525187957422711619, 8.232982112858895263400869753342
