| L(s) = 1 | − 4.62·2-s − 8.38·3-s + 13.3·4-s + 5·5-s + 38.7·6-s + 7·7-s − 24.9·8-s + 43.3·9-s − 23.1·10-s − 30.1·11-s − 112.·12-s + 88.9·13-s − 32.3·14-s − 41.9·15-s + 8.10·16-s − 4.73·17-s − 200.·18-s + 124.·19-s + 66.9·20-s − 58.7·21-s + 139.·22-s + 20.2·23-s + 208.·24-s + 25·25-s − 411.·26-s − 136.·27-s + 93.7·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 1.61·3-s + 1.67·4-s + 0.447·5-s + 2.63·6-s + 0.377·7-s − 1.10·8-s + 1.60·9-s − 0.731·10-s − 0.825·11-s − 2.70·12-s + 1.89·13-s − 0.617·14-s − 0.721·15-s + 0.126·16-s − 0.0675·17-s − 2.62·18-s + 1.50·19-s + 0.748·20-s − 0.610·21-s + 1.34·22-s + 0.183·23-s + 1.77·24-s + 0.200·25-s − 3.10·26-s − 0.976·27-s + 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4054847925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4054847925\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 4.62T + 8T^{2} \) |
| 3 | \( 1 + 8.38T + 27T^{2} \) |
| 11 | \( 1 + 30.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 88.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.73T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 20.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.03T + 2.97e4T^{2} \) |
| 37 | \( 1 + 141.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 838.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 389.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 697.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 66.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 526.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 70.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 9.27T + 7.04e5T^{2} \) |
| 97 | \( 1 + 4.19T + 9.12e5T^{2} \) |
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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
−16.30505966109582821774926154834, −15.85189850235758509320536365784, −13.41669853626646070067176697869, −11.68393347563111051566319877659, −10.89130227704390868857615237865, −10.00712899667892935679610843811, −8.408801047768907953737951766220, −6.83264824669928917182084762348, −5.47470118189771764634654008489, −1.06945212547537925642012809733,
1.06945212547537925642012809733, 5.47470118189771764634654008489, 6.83264824669928917182084762348, 8.408801047768907953737951766220, 10.00712899667892935679610843811, 10.89130227704390868857615237865, 11.68393347563111051566319877659, 13.41669853626646070067176697869, 15.85189850235758509320536365784, 16.30505966109582821774926154834
