| L(s) = 1 | + 0.261·2-s − 1.21·3-s − 0.931·4-s − 5-s − 0.317·6-s + 1.98·7-s − 0.504·8-s + 0.482·9-s − 0.261·10-s + 1.13·12-s − 1.58·13-s + 0.517·14-s + 1.21·15-s + 0.800·16-s − 1.84·17-s + 0.125·18-s + 19-s + 0.931·20-s − 2.41·21-s + 0.614·24-s + 25-s − 0.414·26-s + 0.630·27-s − 1.84·28-s + 0.317·30-s + 0.713·32-s − 0.482·34-s + ⋯ |
| L(s) = 1 | + 0.261·2-s − 1.21·3-s − 0.931·4-s − 5-s − 0.317·6-s + 1.98·7-s − 0.504·8-s + 0.482·9-s − 0.261·10-s + 1.13·12-s − 1.58·13-s + 0.517·14-s + 1.21·15-s + 0.800·16-s − 1.84·17-s + 0.125·18-s + 19-s + 0.931·20-s − 2.41·21-s + 0.614·24-s + 25-s − 0.414·26-s + 0.630·27-s − 1.84·28-s + 0.317·30-s + 0.713·32-s − 0.482·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1895 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1895 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5657678894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5657678894\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 379 | \( 1 + T \) |
| good | 2 | \( 1 - 0.261T + T^{2} \) |
| 3 | \( 1 + 1.21T + T^{2} \) |
| 7 | \( 1 - 1.98T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.58T + T^{2} \) |
| 17 | \( 1 + 1.84T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.73T + T^{2} \) |
| 43 | \( 1 - 1.58T + T^{2} \) |
| 47 | \( 1 - 0.765T + T^{2} \) |
| 53 | \( 1 + 0.261T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.765T + T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.249680560734960057081201858056, −8.622667749146220817808013703975, −7.68490627881981807969166154497, −7.26701571057124231857616604364, −5.92501461428499639580563230843, −5.03777832890170172006065464898, −4.69159345070295370913734206037, −4.14946630814794770824518423423, −2.47720062712966785771564206536, −0.77226494634370062607844304104,
0.77226494634370062607844304104, 2.47720062712966785771564206536, 4.14946630814794770824518423423, 4.69159345070295370913734206037, 5.03777832890170172006065464898, 5.92501461428499639580563230843, 7.26701571057124231857616604364, 7.68490627881981807969166154497, 8.622667749146220817808013703975, 9.249680560734960057081201858056
