| L(s) = 1 | + 1.33·2-s − 0.223·4-s + 5-s + 4.48·7-s − 2.96·8-s + 1.33·10-s − 5.35·11-s + 0.866·13-s + 5.97·14-s − 3.50·16-s − 3.12·17-s + 2.28·19-s − 0.223·20-s − 7.13·22-s + 8.49·23-s + 25-s + 1.15·26-s − 0.999·28-s − 0.661·29-s + 2.27·31-s + 1.25·32-s − 4.15·34-s + 4.48·35-s − 4.58·37-s + 3.04·38-s − 2.96·40-s + 9.75·41-s + ⋯ |
| L(s) = 1 | + 0.942·2-s − 0.111·4-s + 0.447·5-s + 1.69·7-s − 1.04·8-s + 0.421·10-s − 1.61·11-s + 0.240·13-s + 1.59·14-s − 0.876·16-s − 0.756·17-s + 0.524·19-s − 0.0498·20-s − 1.52·22-s + 1.77·23-s + 0.200·25-s + 0.226·26-s − 0.188·28-s − 0.122·29-s + 0.408·31-s + 0.222·32-s − 0.713·34-s + 0.757·35-s − 0.754·37-s + 0.494·38-s − 0.468·40-s + 1.52·41-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
| \(L(1)\) |
\(\approx\) |
\(3.314562699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.314562699\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.33T + 2T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 11 | \( 1 + 5.35T + 11T^{2} \) |
| 13 | \( 1 - 0.866T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 29 | \( 1 + 0.661T + 29T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 - 9.75T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 - 6.93T + 53T^{2} \) |
| 59 | \( 1 - 8.00T + 59T^{2} \) |
| 61 | \( 1 - 7.59T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 7.95T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 97 | \( 1 - 9.57T + 97T^{2} \) |
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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
−8.619588395081771959012489810330, −7.62566608707357703241967543075, −7.03088688512049458535228776861, −5.69566346957663868921405674792, −5.44498931608359192769609263361, −4.73270664765985050767165802475, −4.17419806906361471742965951063, −2.87915462910578685028186043995, −2.30205036254344130738589331440, −0.932216223507962353797444353044,
0.932216223507962353797444353044, 2.30205036254344130738589331440, 2.87915462910578685028186043995, 4.17419806906361471742965951063, 4.73270664765985050767165802475, 5.44498931608359192769609263361, 5.69566346957663868921405674792, 7.03088688512049458535228776861, 7.62566608707357703241967543075, 8.619588395081771959012489810330
