| L(s) = 1 | + 3.52·2-s − 3·3-s + 4.41·4-s + 17.5·5-s − 10.5·6-s + 32.5·7-s − 12.6·8-s + 9·9-s + 61.9·10-s − 11·11-s − 13.2·12-s − 79.1·13-s + 114.·14-s − 52.7·15-s − 79.8·16-s + 47.0·17-s + 31.7·18-s + 163.·19-s + 77.6·20-s − 97.7·21-s − 38.7·22-s + 69.1·23-s + 37.9·24-s + 184.·25-s − 278.·26-s − 27·27-s + 143.·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 0.577·3-s + 0.551·4-s + 1.57·5-s − 0.719·6-s + 1.75·7-s − 0.558·8-s + 0.333·9-s + 1.96·10-s − 0.301·11-s − 0.318·12-s − 1.68·13-s + 2.19·14-s − 0.908·15-s − 1.24·16-s + 0.670·17-s + 0.415·18-s + 1.97·19-s + 0.867·20-s − 1.01·21-s − 0.375·22-s + 0.626·23-s + 0.322·24-s + 1.47·25-s − 2.10·26-s − 0.192·27-s + 0.970·28-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
| \(L(2)\) |
\(\approx\) |
\(5.772152833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.772152833\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.52T + 8T^{2} \) |
| 5 | \( 1 - 17.5T + 125T^{2} \) |
| 7 | \( 1 - 32.5T + 343T^{2} \) |
| 13 | \( 1 + 79.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 163.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 98.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 253.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 72.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 478.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 423.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 326.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 291.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 288.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 57.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 446.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 361.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 326.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 254.T + 9.12e5T^{2} \) |
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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
−9.137891666645632864283703843924, −7.52748469365286211423429233127, −7.28318946420691804178599195903, −5.72869641237906739592549231832, −5.50094515971179524737636777591, −5.07328633280625572102731064364, −4.23858973680068729960214035649, −2.80091789014938958217049456174, −2.06135198980074087073104461346, −0.991535718753581145180086179627,
0.991535718753581145180086179627, 2.06135198980074087073104461346, 2.80091789014938958217049456174, 4.23858973680068729960214035649, 5.07328633280625572102731064364, 5.50094515971179524737636777591, 5.72869641237906739592549231832, 7.28318946420691804178599195903, 7.52748469365286211423429233127, 9.137891666645632864283703843924
