| L(s) = 1 | + 4.07·2-s − 3·3-s + 8.64·4-s − 17.0·5-s − 12.2·6-s − 11.0·7-s + 2.61·8-s + 9·9-s − 69.3·10-s − 14.9·11-s − 25.9·12-s − 44.8·13-s − 45.1·14-s + 51.0·15-s − 58.4·16-s − 116.·17-s + 36.7·18-s + 49.4·19-s − 146.·20-s + 33.1·21-s − 60.9·22-s − 23·23-s − 7.85·24-s + 164.·25-s − 183.·26-s − 27·27-s − 95.5·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s − 0.577·3-s + 1.08·4-s − 1.52·5-s − 0.832·6-s − 0.597·7-s + 0.115·8-s + 0.333·9-s − 2.19·10-s − 0.409·11-s − 0.623·12-s − 0.957·13-s − 0.861·14-s + 0.878·15-s − 0.913·16-s − 1.65·17-s + 0.480·18-s + 0.596·19-s − 1.64·20-s + 0.344·21-s − 0.590·22-s − 0.208·23-s − 0.0667·24-s + 1.31·25-s − 1.38·26-s − 0.192·27-s − 0.645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5582134198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5582134198\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 23 | \( 1 + 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 4.07T + 8T^{2} \) |
| 5 | \( 1 + 17.0T + 125T^{2} \) |
| 7 | \( 1 + 11.0T + 343T^{2} \) |
| 11 | \( 1 + 14.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.4T + 6.85e3T^{2} \) |
| 31 | \( 1 - 44.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 77.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 546.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 648.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 294.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 346.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 428.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 617.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 499.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 179.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 531.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 980.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 519.T + 9.12e5T^{2} \) |
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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
−8.667529960217584831327315863575, −7.75993388492811979453629719186, −6.83775776519045909006507185954, −6.52840921009551990493997947607, −5.19283549159141129870932544895, −4.82935320276655704669389714080, −3.93467549585326416533584258237, −3.31576106442247666809453095619, −2.27113521423586051145031486692, −0.26310327687657644730790358439,
0.26310327687657644730790358439, 2.27113521423586051145031486692, 3.31576106442247666809453095619, 3.93467549585326416533584258237, 4.82935320276655704669389714080, 5.19283549159141129870932544895, 6.52840921009551990493997947607, 6.83775776519045909006507185954, 7.75993388492811979453629719186, 8.667529960217584831327315863575
