| L(s) = 1 | − 5.24·2-s − 3·3-s + 19.4·4-s + 5·5-s + 15.7·6-s − 18.4·7-s − 60.0·8-s + 9·9-s − 26.2·10-s + 11.5·11-s − 58.3·12-s − 34.4·13-s + 96.6·14-s − 15·15-s + 159.·16-s + 70.7·17-s − 47.1·18-s + 3.52·19-s + 97.3·20-s + 55.3·21-s − 60.4·22-s − 18.3·23-s + 180.·24-s + 25·25-s + 180.·26-s − 27·27-s − 359.·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.577·3-s + 2.43·4-s + 0.447·5-s + 1.06·6-s − 0.996·7-s − 2.65·8-s + 0.333·9-s − 0.828·10-s + 0.316·11-s − 1.40·12-s − 0.735·13-s + 1.84·14-s − 0.258·15-s + 2.48·16-s + 1.00·17-s − 0.617·18-s + 0.0425·19-s + 1.08·20-s + 0.575·21-s − 0.585·22-s − 0.166·23-s + 1.53·24-s + 0.200·25-s + 1.36·26-s − 0.192·27-s − 2.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4745704679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4745704679\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 5.24T + 8T^{2} \) |
| 7 | \( 1 + 18.4T + 343T^{2} \) |
| 11 | \( 1 - 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 3.52T + 6.85e3T^{2} \) |
| 23 | \( 1 + 18.3T + 1.21e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 74.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 405.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 120.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 34.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 873.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 432.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 548.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 511.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 788.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 967.T + 9.12e5T^{2} \) |
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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
−10.24675869339326136148409783891, −9.896558716153945624491933485478, −9.169482857201921686115886014405, −8.092451644929153711759681729775, −7.05607991958623523050390139918, −6.46555914078030966914600100349, −5.39039228381201979468547429343, −3.29659420375294180500554826932, −1.91878660262277380461491291636, −0.58552384414619822930587158013,
0.58552384414619822930587158013, 1.91878660262277380461491291636, 3.29659420375294180500554826932, 5.39039228381201979468547429343, 6.46555914078030966914600100349, 7.05607991958623523050390139918, 8.092451644929153711759681729775, 9.169482857201921686115886014405, 9.896558716153945624491933485478, 10.24675869339326136148409783891
