| L(s) = 1 | + 8.91·2-s + 13.7·3-s + 47.5·4-s − 25·5-s + 122.·6-s + 49·7-s + 138.·8-s − 53.0·9-s − 222.·10-s − 187.·11-s + 655.·12-s + 255.·13-s + 437.·14-s − 344.·15-s − 283.·16-s − 1.10e3·17-s − 473.·18-s + 2.27e3·19-s − 1.18e3·20-s + 675.·21-s − 1.67e3·22-s − 475.·23-s + 1.91e3·24-s + 625·25-s + 2.28e3·26-s − 4.08e3·27-s + 2.33e3·28-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 0.884·3-s + 1.48·4-s − 0.447·5-s + 1.39·6-s + 0.377·7-s + 0.767·8-s − 0.218·9-s − 0.705·10-s − 0.468·11-s + 1.31·12-s + 0.419·13-s + 0.595·14-s − 0.395·15-s − 0.276·16-s − 0.926·17-s − 0.344·18-s + 1.44·19-s − 0.664·20-s + 0.334·21-s − 0.738·22-s − 0.187·23-s + 0.678·24-s + 0.200·25-s + 0.661·26-s − 1.07·27-s + 0.561·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.967358050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.967358050\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 - 8.91T + 32T^{2} \) |
| 3 | \( 1 - 13.7T + 243T^{2} \) |
| 11 | \( 1 + 187.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 255.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 475.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.25e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.39e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.25e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.61e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.92e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.27e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e4T + 8.58e9T^{2} \) |
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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
−15.22300823653396129235520446996, −14.10123988759601133171469416168, −13.50478003827753772307817116974, −12.14326984075404618029475103325, −11.05028075719653888740093359956, −8.910048728507906840925167122442, −7.43451404251766750846190762537, −5.58565587756984507262052215154, −4.02159212380100377408792884699, −2.66426719738050585453327304590,
2.66426719738050585453327304590, 4.02159212380100377408792884699, 5.58565587756984507262052215154, 7.43451404251766750846190762537, 8.910048728507906840925167122442, 11.05028075719653888740093359956, 12.14326984075404618029475103325, 13.50478003827753772307817116974, 14.10123988759601133171469416168, 15.22300823653396129235520446996
