| L(s) = 1 | − 2-s − 1.81·3-s + 4-s + 0.208·5-s + 1.81·6-s − 7-s − 8-s + 0.288·9-s − 0.208·10-s − 2.47·11-s − 1.81·12-s + 5.36·13-s + 14-s − 0.378·15-s + 16-s − 5.49·17-s − 0.288·18-s + 0.208·20-s + 1.81·21-s + 2.47·22-s + 4.71·23-s + 1.81·24-s − 4.95·25-s − 5.36·26-s + 4.91·27-s − 28-s + 0.0880·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.04·3-s + 0.5·4-s + 0.0933·5-s + 0.740·6-s − 0.377·7-s − 0.353·8-s + 0.0961·9-s − 0.0660·10-s − 0.746·11-s − 0.523·12-s + 1.48·13-s + 0.267·14-s − 0.0977·15-s + 0.250·16-s − 1.33·17-s − 0.0679·18-s + 0.0466·20-s + 0.395·21-s + 0.527·22-s + 0.983·23-s + 0.370·24-s − 0.991·25-s − 1.05·26-s + 0.946·27-s − 0.188·28-s + 0.0163·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5132368427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5132368427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.81T + 3T^{2} \) |
| 5 | \( 1 - 0.208T + 5T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 - 0.0880T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 7.36T + 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 + 3.32T + 59T^{2} \) |
| 61 | \( 1 - 0.448T + 61T^{2} \) |
| 67 | \( 1 - 0.511T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 + 1.55T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.378945475625031196303082422016, −7.42171967715474377400631291382, −6.77319331696866879914388076790, −6.01439487172387056847863032683, −5.66188706570108960035654964969, −4.68919864983717871915212155068, −3.68687700280419506927509860099, −2.71979228092010718142000782035, −1.64055078755617400521532312581, −0.45728890021209088225564026246,
0.45728890021209088225564026246, 1.64055078755617400521532312581, 2.71979228092010718142000782035, 3.68687700280419506927509860099, 4.68919864983717871915212155068, 5.66188706570108960035654964969, 6.01439487172387056847863032683, 6.77319331696866879914388076790, 7.42171967715474377400631291382, 8.378945475625031196303082422016
