L(s) = 1 | + 5.18·2-s − 3·3-s + 18.9·4-s − 15.5·6-s + 7·7-s + 56.5·8-s + 9·9-s − 35.9·11-s − 56.7·12-s + 45.2·13-s + 36.3·14-s + 142.·16-s + 113.·17-s + 46.6·18-s + 61.5·19-s − 21·21-s − 186.·22-s − 30.6·23-s − 169.·24-s + 234.·26-s − 27·27-s + 132.·28-s + 214.·29-s + 164.·31-s + 284.·32-s + 107.·33-s + 586.·34-s + ⋯ |
L(s) = 1 | + 1.83·2-s − 0.577·3-s + 2.36·4-s − 1.05·6-s + 0.377·7-s + 2.49·8-s + 0.333·9-s − 0.985·11-s − 1.36·12-s + 0.965·13-s + 0.693·14-s + 2.21·16-s + 1.61·17-s + 0.611·18-s + 0.743·19-s − 0.218·21-s − 1.80·22-s − 0.277·23-s − 1.44·24-s + 1.77·26-s − 0.192·27-s + 0.892·28-s + 1.37·29-s + 0.951·31-s + 1.57·32-s + 0.569·33-s + 2.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.745720871\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.745720871\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
good | 2 | \( 1 - 5.18T + 8T^{2} \) |
| 11 | \( 1 + 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 29.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 151.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 89.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 714.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 297.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 492.T + 9.12e5T^{2} \) |
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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
−10.66680134364680399997145407455, −10.17389922547507373507388901256, −8.315578571604844855612737382642, −7.37911131778149532150762857409, −6.38587890807026824004705173929, −5.47779618243498133099448394630, −5.00309680217051552675187591869, −3.79313639410754705619385576345, −2.85234497190941740337044615919, −1.32330888947937907680771901432,
1.32330888947937907680771901432, 2.85234497190941740337044615919, 3.79313639410754705619385576345, 5.00309680217051552675187591869, 5.47779618243498133099448394630, 6.38587890807026824004705173929, 7.37911131778149532150762857409, 8.315578571604844855612737382642, 10.17389922547507373507388901256, 10.66680134364680399997145407455