| L(s) = 1 | + 5.58·2-s + 3·3-s + 23.1·4-s + 16.7·6-s − 34.4·7-s + 84.4·8-s + 9·9-s + 11·11-s + 69.4·12-s + 71.2·13-s − 192.·14-s + 286.·16-s + 22.3·17-s + 50.2·18-s + 88.1·19-s − 103.·21-s + 61.3·22-s − 21.5·23-s + 253.·24-s + 397.·26-s + 27·27-s − 796.·28-s + 118.·29-s − 33.5·31-s + 922.·32-s + 33·33-s + 124.·34-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 0.577·3-s + 2.89·4-s + 1.13·6-s − 1.85·7-s + 3.73·8-s + 0.333·9-s + 0.301·11-s + 1.67·12-s + 1.52·13-s − 3.66·14-s + 4.47·16-s + 0.318·17-s + 0.657·18-s + 1.06·19-s − 1.07·21-s + 0.594·22-s − 0.195·23-s + 2.15·24-s + 3.00·26-s + 0.192·27-s − 5.37·28-s + 0.757·29-s − 0.194·31-s + 5.09·32-s + 0.174·33-s + 0.629·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.088854555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.088854555\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 5.58T + 8T^{2} \) |
| 7 | \( 1 + 34.4T + 343T^{2} \) |
| 13 | \( 1 - 71.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 33.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 364.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 48.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 95.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 132.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 654.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 772.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 112.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 548.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 559.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 48.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 447.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 552.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 413.T + 9.12e5T^{2} \) |
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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
−10.09966014953495755441857686311, −9.056757366968263878261456928233, −7.74256365548481028103225887644, −6.76575826403278678320897257420, −6.27329950110788387566272738420, −5.44034132340305737539724223403, −4.11283913778136918494513317943, −3.36994724203826607491381042853, −2.95431105892145030442667373624, −1.42547244567983332038333471297,
1.42547244567983332038333471297, 2.95431105892145030442667373624, 3.36994724203826607491381042853, 4.11283913778136918494513317943, 5.44034132340305737539724223403, 6.27329950110788387566272738420, 6.76575826403278678320897257420, 7.74256365548481028103225887644, 9.056757366968263878261456928233, 10.09966014953495755441857686311
