| L(s) = 1 | − 2-s + 2.50·3-s + 4-s + 1.44·5-s − 2.50·6-s − 2.73·7-s − 8-s + 3.28·9-s − 1.44·10-s − 11-s + 2.50·12-s − 0.891·13-s + 2.73·14-s + 3.62·15-s + 16-s + 2.79·17-s − 3.28·18-s − 6.47·19-s + 1.44·20-s − 6.86·21-s + 22-s − 1.36·23-s − 2.50·24-s − 2.91·25-s + 0.891·26-s + 0.724·27-s − 2.73·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.44·3-s + 0.5·4-s + 0.645·5-s − 1.02·6-s − 1.03·7-s − 0.353·8-s + 1.09·9-s − 0.456·10-s − 0.301·11-s + 0.723·12-s − 0.247·13-s + 0.731·14-s + 0.934·15-s + 0.250·16-s + 0.676·17-s − 0.775·18-s − 1.48·19-s + 0.322·20-s − 1.49·21-s + 0.213·22-s − 0.285·23-s − 0.511·24-s − 0.583·25-s + 0.174·26-s + 0.139·27-s − 0.517·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 13 | \( 1 + 0.891T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 0.874T + 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 + 6.67T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 - 0.914T + 61T^{2} \) |
| 67 | \( 1 + 3.25T + 67T^{2} \) |
| 71 | \( 1 + 1.30T + 71T^{2} \) |
| 73 | \( 1 - 9.60T + 73T^{2} \) |
| 79 | \( 1 - 9.86T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 - 7.02T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.180362626672623401946512481445, −7.55460765367212273409821482247, −6.61585454363301534751064627845, −6.14829936055724862478963338732, −5.05664265363220862915279920854, −3.78960770210129613280286475800, −3.20486860620203003629925589751, −2.33628795609618648010321919959, −1.73620374866539679156614081990, 0,
1.73620374866539679156614081990, 2.33628795609618648010321919959, 3.20486860620203003629925589751, 3.78960770210129613280286475800, 5.05664265363220862915279920854, 6.14829936055724862478963338732, 6.61585454363301534751064627845, 7.55460765367212273409821482247, 8.180362626672623401946512481445
