| L(s) = 1 | − 2.27·2-s + 0.848·3-s + 3.18·4-s + 5-s − 1.93·6-s + 1.85·7-s − 2.69·8-s − 2.27·9-s − 2.27·10-s + 4.31·11-s + 2.70·12-s − 4.95·13-s − 4.23·14-s + 0.848·15-s − 0.226·16-s + 17-s + 5.19·18-s − 0.700·19-s + 3.18·20-s + 1.57·21-s − 9.81·22-s − 5.75·23-s − 2.28·24-s + 25-s + 11.2·26-s − 4.48·27-s + 5.91·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.489·3-s + 1.59·4-s + 0.447·5-s − 0.788·6-s + 0.702·7-s − 0.953·8-s − 0.759·9-s − 0.720·10-s + 1.30·11-s + 0.780·12-s − 1.37·13-s − 1.13·14-s + 0.219·15-s − 0.0567·16-s + 0.242·17-s + 1.22·18-s − 0.160·19-s + 0.712·20-s + 0.344·21-s − 2.09·22-s − 1.19·23-s − 0.467·24-s + 0.200·25-s + 2.21·26-s − 0.862·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 0.848T + 3T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 19 | \( 1 + 0.700T + 19T^{2} \) |
| 23 | \( 1 + 5.75T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 + 3.02T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 2.60T + 79T^{2} \) |
| 83 | \( 1 - 3.28T + 83T^{2} \) |
| 89 | \( 1 + 4.05T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−7.82802946472090506713843077746, −7.44863613798279104384866339722, −6.49954342182521442090432245825, −5.87975401815144480708206427499, −4.84089067912959575329552997237, −3.95136544572978457854718689251, −2.72693962745881919587070238204, −2.07699124845237240844070383781, −1.31196897504962072261300345397, 0,
1.31196897504962072261300345397, 2.07699124845237240844070383781, 2.72693962745881919587070238204, 3.95136544572978457854718689251, 4.84089067912959575329552997237, 5.87975401815144480708206427499, 6.49954342182521442090432245825, 7.44863613798279104384866339722, 7.82802946472090506713843077746
