| L(s) = 1 | − 4.00·2-s + 3·3-s + 8.04·4-s − 13.8·5-s − 12.0·6-s + 32.5·7-s − 0.184·8-s + 9·9-s + 55.3·10-s + 11·11-s + 24.1·12-s − 17.4·13-s − 130.·14-s − 41.4·15-s − 63.6·16-s + 109.·17-s − 36.0·18-s − 99.8·19-s − 111.·20-s + 97.5·21-s − 44.0·22-s + 15.9·23-s − 0.552·24-s + 65.9·25-s + 70.0·26-s + 27·27-s + 261.·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 1.00·4-s − 1.23·5-s − 0.817·6-s + 1.75·7-s − 0.00813·8-s + 0.333·9-s + 1.75·10-s + 0.301·11-s + 0.580·12-s − 0.373·13-s − 2.48·14-s − 0.713·15-s − 0.994·16-s + 1.56·17-s − 0.472·18-s − 1.20·19-s − 1.24·20-s + 1.01·21-s − 0.427·22-s + 0.144·23-s − 0.00469·24-s + 0.527·25-s + 0.528·26-s + 0.192·27-s + 1.76·28-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
| \(L(2)\) |
\(\approx\) |
\(1.292184163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292184163\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.00T + 8T^{2} \) |
| 5 | \( 1 + 13.8T + 125T^{2} \) |
| 7 | \( 1 - 32.5T + 343T^{2} \) |
| 13 | \( 1 + 17.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 15.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 56.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 265.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 26.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 270.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 421.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 637.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 617.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 592.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 984.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 182.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 511.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 992.T + 9.12e5T^{2} \) |
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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
−8.589528570770782058144035668836, −8.125965727775149251146051304245, −7.54466708816296469680677735515, −7.18977582169869534962325104300, −5.61533483848395693901026623715, −4.46151501086152785946923407423, −3.97447791457931230166045480045, −2.51640186032725933330919562044, −1.53023530868669491773215970856, −0.68197544013886381754125227803,
0.68197544013886381754125227803, 1.53023530868669491773215970856, 2.51640186032725933330919562044, 3.97447791457931230166045480045, 4.46151501086152785946923407423, 5.61533483848395693901026623715, 7.18977582169869534962325104300, 7.54466708816296469680677735515, 8.125965727775149251146051304245, 8.589528570770782058144035668836
