| L(s) = 1 | − 15·2-s + 161·4-s − 246·5-s − 430·7-s − 1.45e3·8-s + 729·9-s + 3.69e3·10-s + 6.45e3·14-s + 1.15e4·16-s − 1.09e4·18-s + 1.06e4·19-s − 3.96e4·20-s + 4.48e4·25-s − 6.92e4·28-s − 2.97e4·31-s − 7.96e4·32-s + 1.05e5·35-s + 1.17e5·36-s − 1.59e5·38-s + 3.57e5·40-s − 6.05e4·41-s − 1.79e5·45-s + 1.71e5·47-s + 6.72e4·49-s − 6.73e5·50-s + 6.25e5·56-s + 1.36e5·59-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.51·4-s − 1.96·5-s − 1.25·7-s − 2.84·8-s + 9-s + 3.68·10-s + 2.35·14-s + 2.81·16-s − 1.87·18-s + 1.54·19-s − 4.95·20-s + 2.87·25-s − 3.15·28-s − 31-s − 2.43·32-s + 2.46·35-s + 2.51·36-s − 2.90·38-s + 5.59·40-s − 0.878·41-s − 1.96·45-s + 1.65·47-s + 0.571·49-s − 5.38·50-s + 3.56·56-s + 0.666·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3484160247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3484160247\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + p^{3} T \) |
| good | 2 | \( 1 + 15 T + p^{6} T^{2} \) |
| 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( 1 + 246 T + p^{6} T^{2} \) |
| 7 | \( 1 + 430 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 - 10618 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 + 60558 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 - 171810 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( 1 - 136842 T + p^{6} T^{2} \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 + 133670 T + p^{6} T^{2} \) |
| 71 | \( 1 - 284178 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 - 1807490 T + p^{6} T^{2} \) |
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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
−15.94111592159695384775215938261, −15.36165536989464381497875641152, −12.51334466709860948944121497719, −11.53053452625768437971591688568, −10.25959883495607747915136177012, −9.065118761165065536808628316593, −7.64827883225499167030293822542, −6.97167795634074739591920598872, −3.42259970738871566827389033475, −0.65987254774365550354591134882,
0.65987254774365550354591134882, 3.42259970738871566827389033475, 6.97167795634074739591920598872, 7.64827883225499167030293822542, 9.065118761165065536808628316593, 10.25959883495607747915136177012, 11.53053452625768437971591688568, 12.51334466709860948944121497719, 15.36165536989464381497875641152, 15.94111592159695384775215938261
