| L(s) = 1 | − 17.3·2-s − 89.1·3-s + 173.·4-s − 9.67·5-s + 1.54e3·6-s − 343·7-s − 784.·8-s + 5.75e3·9-s + 167.·10-s − 4.81e3·11-s − 1.54e4·12-s − 2.19e3·13-s + 5.95e3·14-s + 862.·15-s − 8.55e3·16-s − 1.32e4·17-s − 9.99e4·18-s − 2.16e4·19-s − 1.67e3·20-s + 3.05e4·21-s + 8.34e4·22-s − 8.74e4·23-s + 6.99e4·24-s − 7.80e4·25-s + 3.81e4·26-s − 3.18e5·27-s − 5.94e4·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 1.90·3-s + 1.35·4-s − 0.0346·5-s + 2.92·6-s − 0.377·7-s − 0.541·8-s + 2.63·9-s + 0.0530·10-s − 1.08·11-s − 2.57·12-s − 0.277·13-s + 0.579·14-s + 0.0659·15-s − 0.521·16-s − 0.652·17-s − 4.03·18-s − 0.725·19-s − 0.0468·20-s + 0.720·21-s + 1.67·22-s − 1.49·23-s + 1.03·24-s − 0.998·25-s + 0.425·26-s − 3.11·27-s − 0.511·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.02788899752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02788899752\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 + 17.3T + 128T^{2} \) |
| 3 | \( 1 + 89.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 9.67T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.81e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.16e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.74e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.95e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.66e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.25e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.97e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.17e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.88e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.45e3T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.64e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.80e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.68e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.15e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.71e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.99e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.25e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.31e6T + 8.07e13T^{2} \) |
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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
−12.20953019503828590342288361296, −11.25584704673034164108744999224, −10.36410195170605598235872847705, −9.840513413058647138061469128417, −8.157462554479898743415817590893, −6.99085379333460602103789231705, −6.02207675142202267993555627569, −4.61872304528981445914895013324, −1.81125503792889124325968887100, −0.14246684585066868369796002863,
0.14246684585066868369796002863, 1.81125503792889124325968887100, 4.61872304528981445914895013324, 6.02207675142202267993555627569, 6.99085379333460602103789231705, 8.157462554479898743415817590893, 9.840513413058647138061469128417, 10.36410195170605598235872847705, 11.25584704673034164108744999224, 12.20953019503828590342288361296
