| L(s) = 1 | + 1.03i·5-s + i·7-s − 0.378·11-s − 2·13-s − 3.86i·17-s − 1.46i·19-s + 5.27·23-s + 3.92·25-s + 3.48i·29-s + 2.53i·31-s − 1.03·35-s − 2.92·37-s − 8.76i·41-s − 4i·43-s + 8.48·47-s + ⋯ |
| L(s) = 1 | + 0.462i·5-s + 0.377i·7-s − 0.114·11-s − 0.554·13-s − 0.937i·17-s − 0.335i·19-s + 1.10·23-s + 0.785·25-s + 0.647i·29-s + 0.455i·31-s − 0.174·35-s − 0.481·37-s − 1.36i·41-s − 0.609i·43-s + 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.782320613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782320613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 1.03iT - 5T^{2} \) |
| 11 | \( 1 + 0.378T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.86iT - 17T^{2} \) |
| 19 | \( 1 + 1.46iT - 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 - 3.48iT - 29T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 + 8.76iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 - 8.53iT - 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 1.03iT - 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−8.766510384600569423081665173946, −7.56883162710215058392712531921, −7.09161994994547724346913131574, −6.48274605090713332317658471260, −5.28212653906210004447371406995, −5.04362240120190141958655693988, −3.83379627289559495344903576464, −2.90453478141764417338611059221, −2.30487202291604788152006065815, −0.856363261104878066327038119864,
0.69281928844263599315628919473, 1.79028862109482385363615671781, 2.88513578359844996642913323979, 3.81686244814187671719908351690, 4.65649576980127419924766004860, 5.26588454475033639900562480208, 6.23016239327370004114484346997, 6.87163671044104591955431833350, 7.79073251573308345553924160161, 8.246060808445495298678879369678
