| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 0.255·7-s − 8-s + 9-s + 2.03·11-s + 12-s − 1.70·13-s + 0.255·14-s + 16-s + 5.83·17-s − 18-s + 6.09·19-s − 0.255·21-s − 2.03·22-s − 4.83·23-s − 24-s + 1.70·26-s + 27-s − 0.255·28-s − 2.58·29-s + 31-s − 32-s + 2.03·33-s − 5.83·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.0964·7-s − 0.353·8-s + 0.333·9-s + 0.614·11-s + 0.288·12-s − 0.473·13-s + 0.0681·14-s + 0.250·16-s + 1.41·17-s − 0.235·18-s + 1.39·19-s − 0.0556·21-s − 0.434·22-s − 1.00·23-s − 0.204·24-s + 0.334·26-s + 0.192·27-s − 0.0482·28-s − 0.479·29-s + 0.179·31-s − 0.176·32-s + 0.354·33-s − 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.892831199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892831199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 0.255T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 0.255T + 43T^{2} \) |
| 47 | \( 1 - 9.83T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 0.160T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 5.64T + 71T^{2} \) |
| 73 | \( 1 + 9.93T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−8.242898848735316158453573376243, −7.56669689675696963760335528713, −7.29674033147775441905689091109, −6.15092448676685777203414294686, −5.58577619942438223765742135691, −4.47307471126445578545825550037, −3.53352004841464667175858389031, −2.86816622706733407063495071947, −1.80231703329858121467233080375, −0.859898393976762891322248239361,
0.859898393976762891322248239361, 1.80231703329858121467233080375, 2.86816622706733407063495071947, 3.53352004841464667175858389031, 4.47307471126445578545825550037, 5.58577619942438223765742135691, 6.15092448676685777203414294686, 7.29674033147775441905689091109, 7.56669689675696963760335528713, 8.242898848735316158453573376243
