| L(s) = 1 | + 0.865·2-s + 9·3-s − 31.2·4-s + 26.2·5-s + 7.79·6-s − 186.·7-s − 54.7·8-s + 81·9-s + 22.7·10-s − 263.·11-s − 281.·12-s + 201.·13-s − 161.·14-s + 235.·15-s + 952.·16-s − 389.·17-s + 70.1·18-s − 1.04e3·19-s − 819.·20-s − 1.68e3·21-s − 228.·22-s + 2.48e3·23-s − 492.·24-s − 2.43e3·25-s + 174.·26-s + 729·27-s + 5.84e3·28-s + ⋯ |
| L(s) = 1 | + 0.153·2-s + 0.577·3-s − 0.976·4-s + 0.469·5-s + 0.0883·6-s − 1.44·7-s − 0.302·8-s + 0.333·9-s + 0.0717·10-s − 0.657·11-s − 0.563·12-s + 0.330·13-s − 0.220·14-s + 0.270·15-s + 0.930·16-s − 0.326·17-s + 0.0510·18-s − 0.661·19-s − 0.458·20-s − 0.832·21-s − 0.100·22-s + 0.978·23-s − 0.174·24-s − 0.779·25-s + 0.0505·26-s + 0.192·27-s + 1.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.629480090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.629480090\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 - 0.865T + 32T^{2} \) |
| 5 | \( 1 - 26.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 186.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 263.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 201.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 389.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.81e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.52e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.54e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.41e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.12e4T + 8.58e9T^{2} \) |
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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
−10.37878521719288345829155006270, −9.922633976879095225921073585928, −8.962684691811768571448341188007, −8.305345192726640474550213317348, −6.86282156024608173501388565276, −5.92265754017472026526831580024, −4.69064517278777262284876917460, −3.53569091481277853316917117819, −2.56676930172828736647476545950, −0.66907925696886976219637000345,
0.66907925696886976219637000345, 2.56676930172828736647476545950, 3.53569091481277853316917117819, 4.69064517278777262284876917460, 5.92265754017472026526831580024, 6.86282156024608173501388565276, 8.305345192726640474550213317348, 8.962684691811768571448341188007, 9.922633976879095225921073585928, 10.37878521719288345829155006270
