| L(s) = 1 | − 1.59·2-s − 5.45·4-s + 2.55·5-s − 7·7-s + 21.4·8-s − 4.07·10-s − 8.09·11-s − 26.2·13-s + 11.1·14-s + 9.41·16-s + 69.7·17-s + 105.·19-s − 13.9·20-s + 12.9·22-s − 154.·23-s − 118.·25-s + 41.8·26-s + 38.1·28-s + 72.6·29-s + 282.·31-s − 186.·32-s − 111.·34-s − 17.8·35-s + 25.7·37-s − 168.·38-s + 54.8·40-s + 87.1·41-s + ⋯ |
| L(s) = 1 | − 0.563·2-s − 0.682·4-s + 0.228·5-s − 0.377·7-s + 0.948·8-s − 0.128·10-s − 0.221·11-s − 0.559·13-s + 0.213·14-s + 0.147·16-s + 0.994·17-s + 1.27·19-s − 0.155·20-s + 0.125·22-s − 1.39·23-s − 0.947·25-s + 0.315·26-s + 0.257·28-s + 0.465·29-s + 1.63·31-s − 1.03·32-s − 0.561·34-s − 0.0864·35-s + 0.114·37-s − 0.720·38-s + 0.216·40-s + 0.331·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 1.59T + 8T^{2} \) |
| 5 | \( 1 - 2.55T + 125T^{2} \) |
| 11 | \( 1 + 8.09T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 69.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 72.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 25.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 87.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 89.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 306.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 746.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 525.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 643.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 309.T + 9.12e5T^{2} \) |
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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
−9.938711373838440658291377652002, −9.215261663272117456862935540573, −8.024186728061772349386067347025, −7.60089934943837821605042583217, −6.19430914565734490934957872803, −5.24570606361666449029164335661, −4.20032568952024024626199123388, −2.94824695488841211449080651444, −1.34716622935420296412785132878, 0,
1.34716622935420296412785132878, 2.94824695488841211449080651444, 4.20032568952024024626199123388, 5.24570606361666449029164335661, 6.19430914565734490934957872803, 7.60089934943837821605042583217, 8.024186728061772349386067347025, 9.215261663272117456862935540573, 9.938711373838440658291377652002
