| L(s) = 1 | + 8.45·2-s + 39.4·4-s − 25·5-s + 22.2·7-s + 63.3·8-s − 211.·10-s − 121·11-s − 225.·13-s + 188.·14-s − 728.·16-s + 1.05e3·17-s + 2.52e3·19-s − 987.·20-s − 1.02e3·22-s + 337.·23-s + 625·25-s − 1.90e3·26-s + 880.·28-s + 7.64e3·29-s + 6.75e3·31-s − 8.18e3·32-s + 8.95e3·34-s − 557.·35-s − 5.66e3·37-s + 2.13e4·38-s − 1.58e3·40-s + 1.33e4·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.23·4-s − 0.447·5-s + 0.171·7-s + 0.349·8-s − 0.668·10-s − 0.301·11-s − 0.370·13-s + 0.257·14-s − 0.711·16-s + 0.889·17-s + 1.60·19-s − 0.551·20-s − 0.450·22-s + 0.133·23-s + 0.200·25-s − 0.553·26-s + 0.212·28-s + 1.68·29-s + 1.26·31-s − 1.41·32-s + 1.32·34-s − 0.0769·35-s − 0.680·37-s + 2.39·38-s − 0.156·40-s + 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.835106025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.835106025\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 8.45T + 32T^{2} \) |
| 7 | \( 1 - 22.2T + 1.68e4T^{2} \) |
| 13 | \( 1 + 225.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.52e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 337.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.00e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.66e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.82e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.38e4T + 8.58e9T^{2} \) |
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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
−10.37117571244943784192254577707, −9.343127027676787152790042846424, −8.074541139201004139163656373183, −7.24925678992377684368541875688, −6.17933242530937389304280238677, −5.21548773514697527245232102110, −4.54117328911243111348877274466, −3.39334906295215263807667033696, −2.64282578545768396081758495353, −0.908247883645748048820377656103,
0.908247883645748048820377656103, 2.64282578545768396081758495353, 3.39334906295215263807667033696, 4.54117328911243111348877274466, 5.21548773514697527245232102110, 6.17933242530937389304280238677, 7.24925678992377684368541875688, 8.074541139201004139163656373183, 9.343127027676787152790042846424, 10.37117571244943784192254577707
