| L(s) = 1 | − 3.88·2-s − 27·3-s − 112.·4-s − 26.2·5-s + 104.·6-s − 644.·7-s + 936.·8-s + 729·9-s + 102.·10-s − 5.69e3·11-s + 3.04e3·12-s − 606.·13-s + 2.50e3·14-s + 709.·15-s + 1.08e4·16-s − 3.92e4·17-s − 2.83e3·18-s − 3.40e4·19-s + 2.96e3·20-s + 1.74e4·21-s + 2.21e4·22-s − 2.12e4·23-s − 2.52e4·24-s − 7.74e4·25-s + 2.35e3·26-s − 1.96e4·27-s + 7.27e4·28-s + ⋯ |
| L(s) = 1 | − 0.343·2-s − 0.577·3-s − 0.882·4-s − 0.0939·5-s + 0.198·6-s − 0.710·7-s + 0.646·8-s + 0.333·9-s + 0.0322·10-s − 1.28·11-s + 0.509·12-s − 0.0765·13-s + 0.243·14-s + 0.0542·15-s + 0.659·16-s − 1.93·17-s − 0.114·18-s − 1.13·19-s + 0.0828·20-s + 0.410·21-s + 0.442·22-s − 0.364·23-s − 0.373·24-s − 0.991·25-s + 0.0262·26-s − 0.192·27-s + 0.626·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.007204235492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007204235492\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 3.88T + 128T^{2} \) |
| 5 | \( 1 + 26.2T + 7.81e4T^{2} \) |
| 7 | \( 1 + 644.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.69e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 606.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.12e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.61e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.10e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.30e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.67e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.38e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 3.54e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.67e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.13e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.04e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.39e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.28e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.46e6T + 8.07e13T^{2} \) |
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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
−11.10229280246720388088614293380, −10.41674829570032797699981070828, −9.402130789358016481188734778215, −8.468442286917397274098503711489, −7.31218007745734562015194751431, −6.06780053545432742656865232633, −4.91924307623674576190623502250, −3.88404465633245262407654514559, −2.11051710880023140051078302191, −0.04779777867369096037625480846,
0.04779777867369096037625480846, 2.11051710880023140051078302191, 3.88404465633245262407654514559, 4.91924307623674576190623502250, 6.06780053545432742656865232633, 7.31218007745734562015194751431, 8.468442286917397274098503711489, 9.402130789358016481188734778215, 10.41674829570032797699981070828, 11.10229280246720388088614293380
