| L(s) = 1 | + 0.274·2-s − 3-s − 1.92·4-s + 5-s − 0.274·6-s + 0.857·7-s − 1.07·8-s + 9-s + 0.274·10-s + 2.26·11-s + 1.92·12-s + 6.47·13-s + 0.235·14-s − 15-s + 3.55·16-s + 5.15·17-s + 0.274·18-s + 4.82·19-s − 1.92·20-s − 0.857·21-s + 0.620·22-s + 6.01·23-s + 1.07·24-s + 25-s + 1.77·26-s − 27-s − 1.65·28-s + ⋯ |
| L(s) = 1 | + 0.193·2-s − 0.577·3-s − 0.962·4-s + 0.447·5-s − 0.111·6-s + 0.324·7-s − 0.380·8-s + 0.333·9-s + 0.0866·10-s + 0.682·11-s + 0.555·12-s + 1.79·13-s + 0.0628·14-s − 0.258·15-s + 0.888·16-s + 1.25·17-s + 0.0646·18-s + 1.10·19-s − 0.430·20-s − 0.187·21-s + 0.132·22-s + 1.25·23-s + 0.219·24-s + 0.200·25-s + 0.348·26-s − 0.192·27-s − 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.264927305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264927305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.274T + 2T^{2} \) |
| 7 | \( 1 - 0.857T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 6.50T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 2.91T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 1.05T + 61T^{2} \) |
| 67 | \( 1 - 7.50T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 5.76T + 83T^{2} \) |
| 89 | \( 1 - 2.61T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.419714123351790872097138917636, −7.30557756161087347294975421513, −6.41710449224115356377742630726, −5.89889291621256944589877373311, −5.15827791904687971898687275352, −4.64893305446254974667588992705, −3.60379849574402220354854963792, −3.15405644860026591485668956973, −1.33776031116108600915015847536, −0.994398942317040382962858979770,
0.994398942317040382962858979770, 1.33776031116108600915015847536, 3.15405644860026591485668956973, 3.60379849574402220354854963792, 4.64893305446254974667588992705, 5.15827791904687971898687275352, 5.89889291621256944589877373311, 6.41710449224115356377742630726, 7.30557756161087347294975421513, 8.419714123351790872097138917636
