L(s) = 1 | − 2-s + 2.78·3-s + 4-s + 2.54·5-s − 2.78·6-s − 7-s − 8-s + 4.75·9-s − 2.54·10-s + 3.33·11-s + 2.78·12-s + 14-s + 7.09·15-s + 16-s + 7.09·17-s − 4.75·18-s − 4.54·19-s + 2.54·20-s − 2.78·21-s − 3.33·22-s + 1.75·23-s − 2.78·24-s + 1.48·25-s + 4.90·27-s − 28-s + 7.57·29-s − 7.09·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.60·3-s + 0.5·4-s + 1.13·5-s − 1.13·6-s − 0.377·7-s − 0.353·8-s + 1.58·9-s − 0.804·10-s + 1.00·11-s + 0.804·12-s + 0.267·14-s + 1.83·15-s + 0.250·16-s + 1.71·17-s − 1.12·18-s − 1.04·19-s + 0.569·20-s − 0.607·21-s − 0.710·22-s + 0.366·23-s − 0.568·24-s + 0.296·25-s + 0.943·27-s − 0.188·28-s + 1.40·29-s − 1.29·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.053872064\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.053872064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 2.78T + 3T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 + 4.54T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 + 3.33T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 + 0.785T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 9.33T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−9.118845120949580933362561171269, −8.312424085690773376734974493654, −7.77196148387797437479645854702, −6.69320041380236837939052127835, −6.24221545747172056595185993836, −5.01819187314938525024100768315, −3.67751743626183808932281260047, −3.06149081863010575768760655536, −2.05897451687618667388423324755, −1.33183753168464819506923358904,
1.33183753168464819506923358904, 2.05897451687618667388423324755, 3.06149081863010575768760655536, 3.67751743626183808932281260047, 5.01819187314938525024100768315, 6.24221545747172056595185993836, 6.69320041380236837939052127835, 7.77196148387797437479645854702, 8.312424085690773376734974493654, 9.118845120949580933362561171269