| L(s) = 1 | − 3.81·2-s + 8.34·3-s + 6.58·4-s + 2.56·5-s − 31.8·6-s − 3.42·7-s + 5.39·8-s + 42.6·9-s − 9.79·10-s − 24.5·11-s + 54.9·12-s − 46.2·13-s + 13.0·14-s + 21.4·15-s − 73.3·16-s + 97.5·17-s − 163.·18-s − 131.·19-s + 16.8·20-s − 28.5·21-s + 93.6·22-s + 175.·23-s + 45.0·24-s − 118.·25-s + 176.·26-s + 130.·27-s − 22.5·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 1.60·3-s + 0.823·4-s + 0.229·5-s − 2.16·6-s − 0.184·7-s + 0.238·8-s + 1.58·9-s − 0.309·10-s − 0.672·11-s + 1.32·12-s − 0.987·13-s + 0.249·14-s + 0.368·15-s − 1.14·16-s + 1.39·17-s − 2.13·18-s − 1.59·19-s + 0.188·20-s − 0.296·21-s + 0.907·22-s + 1.58·23-s + 0.383·24-s − 0.947·25-s + 1.33·26-s + 0.933·27-s − 0.152·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 3.81T + 8T^{2} \) |
| 3 | \( 1 - 8.34T + 27T^{2} \) |
| 5 | \( 1 - 2.56T + 125T^{2} \) |
| 7 | \( 1 + 3.42T + 343T^{2} \) |
| 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 83.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 201.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 373.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 578.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 892.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 2.66T + 3.00e5T^{2} \) |
| 71 | \( 1 - 467.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 481.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 296.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 746.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.72e3T + 9.12e5T^{2} \) |
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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
−8.477341974157166994639654911350, −7.943606385962808013647024758444, −7.44994089989908877556145592511, −6.56203972606327802059043692207, −5.16403590861802549616887963501, −4.14521042248147534634623611255, −2.96597054208219591772978239788, −2.32157548281087653833785398468, −1.36499101276782826890288764270, 0,
1.36499101276782826890288764270, 2.32157548281087653833785398468, 2.96597054208219591772978239788, 4.14521042248147534634623611255, 5.16403590861802549616887963501, 6.56203972606327802059043692207, 7.44994089989908877556145592511, 7.943606385962808013647024758444, 8.477341974157166994639654911350
