| L(s) = 1 | + 1.80·2-s − 25.0·3-s − 28.7·4-s + 25·5-s − 45.2·6-s − 12.8·7-s − 109.·8-s + 383.·9-s + 45.1·10-s − 121·11-s + 719.·12-s − 667.·13-s − 23.2·14-s − 625.·15-s + 720.·16-s − 546.·17-s + 693.·18-s − 361·19-s − 718.·20-s + 321.·21-s − 218.·22-s − 705.·23-s + 2.74e3·24-s + 625·25-s − 1.20e3·26-s − 3.52e3·27-s + 369.·28-s + ⋯ |
| L(s) = 1 | + 0.319·2-s − 1.60·3-s − 0.897·4-s + 0.447·5-s − 0.513·6-s − 0.0991·7-s − 0.606·8-s + 1.57·9-s + 0.142·10-s − 0.301·11-s + 1.44·12-s − 1.09·13-s − 0.0316·14-s − 0.718·15-s + 0.704·16-s − 0.458·17-s + 0.504·18-s − 0.229·19-s − 0.401·20-s + 0.159·21-s − 0.0963·22-s − 0.278·23-s + 0.973·24-s + 0.200·25-s − 0.350·26-s − 0.929·27-s + 0.0889·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1642179589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1642179589\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 1.80T + 32T^{2} \) |
| 3 | \( 1 + 25.0T + 243T^{2} \) |
| 7 | \( 1 + 12.8T + 1.68e4T^{2} \) |
| 13 | \( 1 + 667.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 546.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 705.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.85e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.42e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.69e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.84e4T + 8.58e9T^{2} \) |
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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
−9.560238814143301079070149561606, −8.371930006250275278282784118112, −7.30800628888378475876943700560, −6.28479790308615952045663709718, −5.75548490116336430924529954983, −4.80624578778689234313974340425, −4.49242911071445168613174166355, −2.99295479500397301170559203448, −1.51646628988643258184415419449, −0.18692998725137699895927118540,
0.18692998725137699895927118540, 1.51646628988643258184415419449, 2.99295479500397301170559203448, 4.49242911071445168613174166355, 4.80624578778689234313974340425, 5.75548490116336430924529954983, 6.28479790308615952045663709718, 7.30800628888378475876943700560, 8.371930006250275278282784118112, 9.560238814143301079070149561606
