| L(s) = 1 | + 2-s − 0.585·3-s + 4-s − 0.585·6-s + 8-s − 2.65·9-s + 4.82·11-s − 0.585·12-s − 0.828·13-s + 16-s − 5.41·17-s − 2.65·18-s + 3.41·19-s + 4.82·22-s + 6.82·23-s − 0.585·24-s − 0.828·26-s + 3.31·27-s + 0.828·29-s + 2.82·31-s + 32-s − 2.82·33-s − 5.41·34-s − 2.65·36-s − 3.65·37-s + 3.41·38-s + 0.485·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.338·3-s + 0.5·4-s − 0.239·6-s + 0.353·8-s − 0.885·9-s + 1.45·11-s − 0.169·12-s − 0.229·13-s + 0.250·16-s − 1.31·17-s − 0.626·18-s + 0.783·19-s + 1.02·22-s + 1.42·23-s − 0.119·24-s − 0.162·26-s + 0.637·27-s + 0.153·29-s + 0.508·31-s + 0.176·32-s − 0.492·33-s − 0.928·34-s − 0.442·36-s − 0.601·37-s + 0.553·38-s + 0.0777·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.563580022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.563580022\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 + 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.003555614241187848308917353347, −8.201254523175589357966630569448, −7.00895855375079864722618698888, −6.63541974228189732737278139590, −5.77550573270655507526439868050, −4.99364335565615103537599241554, −4.20516015212238769926970896368, −3.26874178011577849744652129922, −2.34212565232817952324516023042, −0.962890402474903067441645072620,
0.962890402474903067441645072620, 2.34212565232817952324516023042, 3.26874178011577849744652129922, 4.20516015212238769926970896368, 4.99364335565615103537599241554, 5.77550573270655507526439868050, 6.63541974228189732737278139590, 7.00895855375079864722618698888, 8.201254523175589357966630569448, 9.003555614241187848308917353347
