| L(s) = 1 | − 5.32·2-s − 3·3-s + 20.3·4-s + 9.51·5-s + 15.9·6-s + 0.376·7-s − 65.9·8-s + 9·9-s − 50.6·10-s − 11·11-s − 61.1·12-s + 32.7·13-s − 2.00·14-s − 28.5·15-s + 188.·16-s − 41.9·17-s − 47.9·18-s − 110.·19-s + 193.·20-s − 1.13·21-s + 58.6·22-s + 79.5·23-s + 197.·24-s − 34.4·25-s − 174.·26-s − 27·27-s + 7.68·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 0.577·3-s + 2.54·4-s + 0.851·5-s + 1.08·6-s + 0.0203·7-s − 2.91·8-s + 0.333·9-s − 1.60·10-s − 0.301·11-s − 1.47·12-s + 0.699·13-s − 0.0383·14-s − 0.491·15-s + 2.94·16-s − 0.598·17-s − 0.627·18-s − 1.33·19-s + 2.16·20-s − 0.0117·21-s + 0.567·22-s + 0.721·23-s + 1.68·24-s − 0.275·25-s − 1.31·26-s − 0.192·27-s + 0.0518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 5.32T + 8T^{2} \) |
| 5 | \( 1 - 9.51T + 125T^{2} \) |
| 7 | \( 1 - 0.376T + 343T^{2} \) |
| 13 | \( 1 - 32.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 41.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 79.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 95.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 492.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 124.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 495.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 167.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 897.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 51.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 494.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.51e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 135.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.07e3T + 9.12e5T^{2} \) |
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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
−8.777087089621557933524121873434, −7.71769429187437879099531967967, −6.97906808211900928554741151522, −6.19097876496930406932842793107, −5.75611313789163174672610291389, −4.34039615165294900011259039942, −2.77488859115843880932175165518, −1.95327706564730346546014959495, −1.05894794738169081790009198415, 0,
1.05894794738169081790009198415, 1.95327706564730346546014959495, 2.77488859115843880932175165518, 4.34039615165294900011259039942, 5.75611313789163174672610291389, 6.19097876496930406932842793107, 6.97906808211900928554741151522, 7.71769429187437879099531967967, 8.777087089621557933524121873434
