| L(s) = 1 | + 2.82·2-s − 0.0470·4-s − 5·5-s − 7.12·7-s − 22.6·8-s − 14.1·10-s + 11·11-s + 66.8·13-s − 20.0·14-s − 63.6·16-s + 40.9·17-s + 4.85·19-s + 0.235·20-s + 31.0·22-s + 128.·23-s + 25·25-s + 188.·26-s + 0.335·28-s + 224.·29-s − 267.·31-s + 2.13·32-s + 115.·34-s + 35.6·35-s + 418.·37-s + 13.6·38-s + 113.·40-s + 496.·41-s + ⋯ |
| L(s) = 1 | + 0.997·2-s − 0.00588·4-s − 0.447·5-s − 0.384·7-s − 1.00·8-s − 0.445·10-s + 0.301·11-s + 1.42·13-s − 0.383·14-s − 0.994·16-s + 0.583·17-s + 0.0585·19-s + 0.00263·20-s + 0.300·22-s + 1.16·23-s + 0.200·25-s + 1.42·26-s + 0.00226·28-s + 1.43·29-s − 1.54·31-s + 0.0117·32-s + 0.581·34-s + 0.172·35-s + 1.85·37-s + 0.0584·38-s + 0.448·40-s + 1.89·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.627997698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.627997698\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 2.82T + 8T^{2} \) |
| 7 | \( 1 + 7.12T + 343T^{2} \) |
| 13 | \( 1 - 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.85T + 6.85e3T^{2} \) |
| 23 | \( 1 - 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 496.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 90.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 203.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 219.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 585.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 156.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 638.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 223.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 415.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 44.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 809.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 429.T + 9.12e5T^{2} \) |
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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
−10.85609212566196742683817734398, −9.493975704129457595656382946240, −8.834167681530892621321917348174, −7.77303228029152404787664574944, −6.51511963292043746976707037569, −5.82003480506397815143996485927, −4.67016657804336166230312374776, −3.75710170207658300263048062688, −2.95182482709861189267656713869, −0.901689433764295354977161719104,
0.901689433764295354977161719104, 2.95182482709861189267656713869, 3.75710170207658300263048062688, 4.67016657804336166230312374776, 5.82003480506397815143996485927, 6.51511963292043746976707037569, 7.77303228029152404787664574944, 8.834167681530892621321917348174, 9.493975704129457595656382946240, 10.85609212566196742683817734398
