L(s) = 1 | − 3-s − 4.12·5-s − 3.19·7-s + 9-s + 1.71·11-s + 2.16·13-s + 4.12·15-s + 8.20·17-s + 1.86·19-s + 3.19·21-s − 4.86·23-s + 12.0·25-s − 27-s − 0.609·29-s − 8.05·31-s − 1.71·33-s + 13.1·35-s − 5.14·37-s − 2.16·39-s − 11.2·41-s − 4.46·43-s − 4.12·45-s + 1.13·47-s + 3.19·49-s − 8.20·51-s − 5.12·53-s − 7.07·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.84·5-s − 1.20·7-s + 0.333·9-s + 0.517·11-s + 0.600·13-s + 1.06·15-s + 1.99·17-s + 0.427·19-s + 0.696·21-s − 1.01·23-s + 2.40·25-s − 0.192·27-s − 0.113·29-s − 1.44·31-s − 0.298·33-s + 2.22·35-s − 0.845·37-s − 0.346·39-s − 1.75·41-s − 0.680·43-s − 0.615·45-s + 0.165·47-s + 0.455·49-s − 1.14·51-s − 0.704·53-s − 0.954·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5770916180\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5770916180\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 4.12T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 - 1.71T + 11T^{2} \) |
| 13 | \( 1 - 2.16T + 13T^{2} \) |
| 17 | \( 1 - 8.20T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 0.609T + 29T^{2} \) |
| 31 | \( 1 + 8.05T + 31T^{2} \) |
| 37 | \( 1 + 5.14T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 47 | \( 1 - 1.13T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 0.674T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 - 5.87T + 89T^{2} \) |
| 97 | \( 1 + 0.476T + 97T^{2} \) |
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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
−7.87712125679821343258806266871, −6.97951525111493576011835483268, −6.71706971741095091105455079086, −5.68261855889168240394791434161, −5.14240136528515806839976025213, −3.93401778299288398687157731975, −3.64442568468770648688061318207, −3.14673070550187541120814930590, −1.45574003914008345995836368491, −0.40957465925589964260463155393,
0.40957465925589964260463155393, 1.45574003914008345995836368491, 3.14673070550187541120814930590, 3.64442568468770648688061318207, 3.93401778299288398687157731975, 5.14240136528515806839976025213, 5.68261855889168240394791434161, 6.71706971741095091105455079086, 6.97951525111493576011835483268, 7.87712125679821343258806266871