| L(s) = 1 | + 10.1·2-s − 15.2·3-s + 71.1·4-s + 64.0·5-s − 155.·6-s + 91.1·7-s + 397.·8-s − 9.94·9-s + 650.·10-s − 484.·11-s − 1.08e3·12-s − 1.04e3·13-s + 925.·14-s − 978.·15-s + 1.75e3·16-s − 375.·17-s − 100.·18-s + 2.89e3·19-s + 4.55e3·20-s − 1.39e3·21-s − 4.91e3·22-s + 170.·23-s − 6.06e3·24-s + 979.·25-s − 1.06e4·26-s + 3.86e3·27-s + 6.48e3·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 0.979·3-s + 2.22·4-s + 1.14·5-s − 1.75·6-s + 0.703·7-s + 2.19·8-s − 0.0409·9-s + 2.05·10-s − 1.20·11-s − 2.17·12-s − 1.72·13-s + 1.26·14-s − 1.12·15-s + 1.71·16-s − 0.315·17-s − 0.0734·18-s + 1.84·19-s + 2.54·20-s − 0.688·21-s − 2.16·22-s + 0.0670·23-s − 2.14·24-s + 0.313·25-s − 3.09·26-s + 1.01·27-s + 1.56·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.290188599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.290188599\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 841T \) |
| good | 2 | \( 1 - 10.1T + 32T^{2} \) |
| 3 | \( 1 + 15.2T + 243T^{2} \) |
| 5 | \( 1 - 64.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 91.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 484.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 375.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.89e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 170.T + 6.43e6T^{2} \) |
| 31 | \( 1 + 3.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.22e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 585.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.99e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.52e4T + 8.58e9T^{2} \) |
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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
−15.82848932651913362116602315980, −14.49025524038437041304063241306, −13.65222944109773222551918707751, −12.42668415061763673571416007975, −11.46532739610619695400200956940, −10.18500835456490329153482939526, −7.17321896279227698373503696451, −5.43648295009015859701682376586, −5.17053504824405639413665915313, −2.45515087959081429292495258402,
2.45515087959081429292495258402, 5.17053504824405639413665915313, 5.43648295009015859701682376586, 7.17321896279227698373503696451, 10.18500835456490329153482939526, 11.46532739610619695400200956940, 12.42668415061763673571416007975, 13.65222944109773222551918707751, 14.49025524038437041304063241306, 15.82848932651913362116602315980
