| L(s) = 1 | − 0.169·2-s + 3·3-s − 7.97·4-s + 0.479·5-s − 0.509·6-s + 32.7·7-s + 2.71·8-s + 9·9-s − 0.0814·10-s − 11·11-s − 23.9·12-s − 42.6·13-s − 5.54·14-s + 1.43·15-s + 63.3·16-s + 48.6·17-s − 1.52·18-s − 11.4·19-s − 3.82·20-s + 98.1·21-s + 1.86·22-s − 68.1·23-s + 8.13·24-s − 124.·25-s + 7.24·26-s + 27·27-s − 260.·28-s + ⋯ |
| L(s) = 1 | − 0.0599·2-s + 0.577·3-s − 0.996·4-s + 0.0429·5-s − 0.0346·6-s + 1.76·7-s + 0.119·8-s + 0.333·9-s − 0.00257·10-s − 0.301·11-s − 0.575·12-s − 0.910·13-s − 0.105·14-s + 0.0247·15-s + 0.989·16-s + 0.693·17-s − 0.0199·18-s − 0.138·19-s − 0.0427·20-s + 1.01·21-s + 0.0180·22-s − 0.618·23-s + 0.0691·24-s − 0.998·25-s + 0.0546·26-s + 0.192·27-s − 1.75·28-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 | \( 1 + 0.169T + 8T^{2} \) |
| 5 | \( 1 - 0.479T + 125T^{2} \) |
| 7 | \( 1 - 32.7T + 343T^{2} \) |
| 13 | \( 1 + 42.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.67T + 2.43e4T^{2} \) |
| 31 | \( 1 + 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 304.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 431.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 131.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 64.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 780.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 763.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 661.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−8.468677684334591670718143968401, −7.74750631154203164614752872713, −7.28132452940959213156106432277, −5.65532874153876252789174098301, −5.10053366422456469653015389090, −4.36656751150573662252614070164, −3.53949558655106396620765109100, −2.19484441429398991918693567801, −1.39527452253446558111648608667, 0,
1.39527452253446558111648608667, 2.19484441429398991918693567801, 3.53949558655106396620765109100, 4.36656751150573662252614070164, 5.10053366422456469653015389090, 5.65532874153876252789174098301, 7.28132452940959213156106432277, 7.74750631154203164614752872713, 8.468677684334591670718143968401
