| L(s) = 1 | − 4.39·2-s − 7.85·3-s + 11.3·4-s − 5.85·5-s + 34.5·6-s − 12.3·7-s − 14.6·8-s + 34.6·9-s + 25.7·10-s + 50.9·11-s − 88.9·12-s − 72.3·13-s + 54.4·14-s + 45.9·15-s − 26.3·16-s + 72.1·17-s − 152.·18-s − 47.7·19-s − 66.2·20-s + 97.3·21-s − 223.·22-s + 194.·23-s + 114.·24-s − 90.7·25-s + 318.·26-s − 60.4·27-s − 140.·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 1.51·3-s + 1.41·4-s − 0.523·5-s + 2.34·6-s − 0.669·7-s − 0.645·8-s + 1.28·9-s + 0.813·10-s + 1.39·11-s − 2.13·12-s − 1.54·13-s + 1.04·14-s + 0.791·15-s − 0.412·16-s + 1.02·17-s − 1.99·18-s − 0.576·19-s − 0.740·20-s + 1.01·21-s − 2.16·22-s + 1.76·23-s + 0.975·24-s − 0.726·25-s + 2.39·26-s − 0.430·27-s − 0.947·28-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 | \( 1 + 4.39T + 8T^{2} \) |
| 3 | \( 1 + 7.85T + 27T^{2} \) |
| 5 | \( 1 + 5.85T + 125T^{2} \) |
| 7 | \( 1 + 12.3T + 343T^{2} \) |
| 11 | \( 1 - 50.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 47.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 19.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 485.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 20.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 276.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 65.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 330.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 522.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 155.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 543.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 781.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 515.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 901.T + 9.12e5T^{2} \) |
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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
−8.665200512684284606227450038329, −7.53233283708386060379777850853, −7.01573683718324617556567594979, −6.46788560149389683870148435076, −5.42166654687999291677992550602, −4.53566563303804027522704643142, −3.33889835277803756944088079021, −1.78309424546899638141960874866, −0.73995840447653602018316441172, 0,
0.73995840447653602018316441172, 1.78309424546899638141960874866, 3.33889835277803756944088079021, 4.53566563303804027522704643142, 5.42166654687999291677992550602, 6.46788560149389683870148435076, 7.01573683718324617556567594979, 7.53233283708386060379777850853, 8.665200512684284606227450038329
