| L(s) = 1 | − 2-s + 2.41·3-s + 4-s − 2.41·6-s + 1.58·7-s − 8-s + 2.82·9-s + 1.41·11-s + 2.41·12-s + 0.171·13-s − 1.58·14-s + 16-s + 17-s − 2.82·18-s − 19-s + 3.82·21-s − 1.41·22-s + 9.24·23-s − 2.41·24-s − 0.171·26-s − 0.414·27-s + 1.58·28-s − 5.82·29-s − 2.24·31-s − 32-s + 3.41·33-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.39·3-s + 0.5·4-s − 0.985·6-s + 0.599·7-s − 0.353·8-s + 0.942·9-s + 0.426·11-s + 0.696·12-s + 0.0475·13-s − 0.423·14-s + 0.250·16-s + 0.242·17-s − 0.666·18-s − 0.229·19-s + 0.835·21-s − 0.301·22-s + 1.92·23-s − 0.492·24-s − 0.0336·26-s − 0.0797·27-s + 0.299·28-s − 1.08·29-s − 0.402·31-s − 0.176·32-s + 0.594·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.041771915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041771915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 0.171T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 - 9.24T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.48T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−9.485047597210104391000956809384, −9.314120936575675239669110822531, −8.413752961397944145466971390681, −7.74352952259069312063347770358, −7.08470267100953588111221212276, −5.86080638297475991552836630505, −4.54338866634520780302783749167, −3.40785226109579241709062727415, −2.48002232869173389252567368619, −1.34184171380425132232223414114,
1.34184171380425132232223414114, 2.48002232869173389252567368619, 3.40785226109579241709062727415, 4.54338866634520780302783749167, 5.86080638297475991552836630505, 7.08470267100953588111221212276, 7.74352952259069312063347770358, 8.413752961397944145466971390681, 9.314120936575675239669110822531, 9.485047597210104391000956809384
