| L(s) = 1 | + 2.32·2-s + 3-s + 3.40·4-s + 3.18·5-s + 2.32·6-s − 7-s + 3.27·8-s + 9-s + 7.40·10-s − 1.26·11-s + 3.40·12-s + 5.18·13-s − 2.32·14-s + 3.18·15-s + 0.795·16-s − 7.01·17-s + 2.32·18-s + 1.76·19-s + 10.8·20-s − 21-s − 2.94·22-s + 8.99·23-s + 3.27·24-s + 5.12·25-s + 12.0·26-s + 27-s − 3.40·28-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 0.577·3-s + 1.70·4-s + 1.42·5-s + 0.949·6-s − 0.377·7-s + 1.15·8-s + 0.333·9-s + 2.34·10-s − 0.382·11-s + 0.983·12-s + 1.43·13-s − 0.621·14-s + 0.821·15-s + 0.198·16-s − 1.70·17-s + 0.548·18-s + 0.404·19-s + 2.42·20-s − 0.218·21-s − 0.628·22-s + 1.87·23-s + 0.668·24-s + 1.02·25-s + 2.36·26-s + 0.192·27-s − 0.643·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.277420941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.277420941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.32T + 2T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 + 7.01T + 17T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 - 8.99T + 23T^{2} \) |
| 29 | \( 1 + 9.96T + 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 - 8.09T + 37T^{2} \) |
| 41 | \( 1 + 8.00T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 8.68T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 6.11T + 61T^{2} \) |
| 67 | \( 1 + 3.96T + 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 + 2.06T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 7.94T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 6.71T + 97T^{2} \) |
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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
−7.49633996314115250754758213188, −6.73723090486541635868544132271, −6.35894502388051292593643530392, −5.56521251716726933230535097764, −5.19125394875368447005821574404, −4.16735401038371244390533350637, −3.64197892195144432120441254103, −2.62006409142776866301237944870, −2.35008707496496427763996377547, −1.24426649884507313594437727628,
1.24426649884507313594437727628, 2.35008707496496427763996377547, 2.62006409142776866301237944870, 3.64197892195144432120441254103, 4.16735401038371244390533350637, 5.19125394875368447005821574404, 5.56521251716726933230535097764, 6.35894502388051292593643530392, 6.73723090486541635868544132271, 7.49633996314115250754758213188
