| L(s) = 1 | − 2.23·2-s + 1.75·3-s − 3.01·4-s + 8.98·5-s − 3.92·6-s + 24.5·8-s − 23.9·9-s − 20.0·10-s − 10.1·11-s − 5.30·12-s − 8.62·13-s + 15.8·15-s − 30.7·16-s + 30.9·17-s + 53.3·18-s − 13.8·19-s − 27.1·20-s + 22.7·22-s + 105.·23-s + 43.2·24-s − 44.2·25-s + 19.2·26-s − 89.5·27-s − 196.·29-s − 35.2·30-s + 301.·31-s − 128.·32-s + ⋯ |
| L(s) = 1 | − 0.789·2-s + 0.338·3-s − 0.377·4-s + 0.803·5-s − 0.267·6-s + 1.08·8-s − 0.885·9-s − 0.634·10-s − 0.278·11-s − 0.127·12-s − 0.183·13-s + 0.272·15-s − 0.480·16-s + 0.441·17-s + 0.698·18-s − 0.166·19-s − 0.303·20-s + 0.220·22-s + 0.956·23-s + 0.367·24-s − 0.354·25-s + 0.145·26-s − 0.638·27-s − 1.25·29-s − 0.214·30-s + 1.74·31-s − 0.707·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.225185894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.225185894\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 2.23T + 8T^{2} \) |
| 3 | \( 1 - 1.75T + 27T^{2} \) |
| 5 | \( 1 - 8.98T + 125T^{2} \) |
| 11 | \( 1 + 10.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.62T + 2.19e3T^{2} \) |
| 17 | \( 1 - 30.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 52.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 413.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 160.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 206.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 488.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 753.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 432.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 61.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 29.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 154.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 639.T + 9.12e5T^{2} \) |
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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
−8.665600023906815284431402264950, −8.128063598160033802711047744522, −7.38269730519334407872779759301, −6.35051372074231034243383654534, −5.47798714152792419032465936564, −4.82396328740365053830159069842, −3.64008203276177769925856137996, −2.63013742574587688398820131427, −1.69035259701030624158551306505, −0.55529907361967445872298206297,
0.55529907361967445872298206297, 1.69035259701030624158551306505, 2.63013742574587688398820131427, 3.64008203276177769925856137996, 4.82396328740365053830159069842, 5.47798714152792419032465936564, 6.35051372074231034243383654534, 7.38269730519334407872779759301, 8.128063598160033802711047744522, 8.665600023906815284431402264950
