| L(s) = 1 | + 1.44·2-s − 3·3-s − 5.92·4-s − 17.1·5-s − 4.32·6-s − 35.8·7-s − 20.0·8-s + 9·9-s − 24.6·10-s − 11·11-s + 17.7·12-s − 18.8·13-s − 51.6·14-s + 51.4·15-s + 18.4·16-s + 115.·17-s + 12.9·18-s − 106.·19-s + 101.·20-s + 107.·21-s − 15.8·22-s + 53.7·23-s + 60.1·24-s + 168.·25-s − 27.1·26-s − 27·27-s + 212.·28-s + ⋯ |
| L(s) = 1 | + 0.509·2-s − 0.577·3-s − 0.740·4-s − 1.53·5-s − 0.294·6-s − 1.93·7-s − 0.886·8-s + 0.333·9-s − 0.780·10-s − 0.301·11-s + 0.427·12-s − 0.402·13-s − 0.986·14-s + 0.884·15-s + 0.288·16-s + 1.64·17-s + 0.169·18-s − 1.28·19-s + 1.13·20-s + 1.11·21-s − 0.153·22-s + 0.487·23-s + 0.511·24-s + 1.34·25-s − 0.204·26-s − 0.192·27-s + 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 1.44T + 8T^{2} \) |
| 5 | \( 1 + 17.1T + 125T^{2} \) |
| 7 | \( 1 + 35.8T + 343T^{2} \) |
| 13 | \( 1 + 18.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 36.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 83.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 78.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 483.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 426.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 216.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 586.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 923.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 171.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 368.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 950.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 768.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 90.8T + 9.12e5T^{2} \) |
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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
−8.374323649925868843898545728318, −7.53302298390584205751872144854, −6.73985949297596053897849805123, −5.95910973558057392137907271470, −5.11668329454381570976340742970, −4.17245166564631192495348324918, −3.55253340319581281311027936104, −2.93242582132244031255993399467, −0.65193354960269550813050834849, 0,
0.65193354960269550813050834849, 2.93242582132244031255993399467, 3.55253340319581281311027936104, 4.17245166564631192495348324918, 5.11668329454381570976340742970, 5.95910973558057392137907271470, 6.73985949297596053897849805123, 7.53302298390584205751872144854, 8.374323649925868843898545728318
