| L(s) = 1 | + 11.1·2-s + 28.1·3-s + 92.7·4-s + 314.·6-s − 135.·7-s + 678.·8-s + 550.·9-s − 74.1·11-s + 2.61e3·12-s + 252.·13-s − 1.51e3·14-s + 4.60e3·16-s − 233.·17-s + 6.15e3·18-s + 550.·19-s − 3.82e3·21-s − 827.·22-s − 1.95e3·23-s + 1.91e4·24-s + 2.81e3·26-s + 8.67e3·27-s − 1.25e4·28-s − 4.67e3·29-s + 3.33e3·31-s + 2.97e4·32-s − 2.08e3·33-s − 2.60e3·34-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 1.80·3-s + 2.89·4-s + 3.56·6-s − 1.04·7-s + 3.74·8-s + 2.26·9-s − 0.184·11-s + 5.23·12-s + 0.414·13-s − 2.06·14-s + 4.49·16-s − 0.195·17-s + 4.47·18-s + 0.349·19-s − 1.89·21-s − 0.364·22-s − 0.769·23-s + 6.77·24-s + 0.817·26-s + 2.29·27-s − 3.03·28-s − 1.03·29-s + 0.622·31-s + 5.13·32-s − 0.333·33-s − 0.386·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(18.05937343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.05937343\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 - 11.1T + 32T^{2} \) |
| 3 | \( 1 - 28.1T + 243T^{2} \) |
| 7 | \( 1 + 135.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 74.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 252.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 233.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 550.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.20e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.91e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.26e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.23e4T + 8.58e9T^{2} \) |
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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
−9.205304305407575833315372102956, −7.948094450817424305881566572464, −7.41827881332084189538318303236, −6.49269437412168924571195937684, −5.73060171033419482492981785043, −4.34191286536892455889759876690, −3.87322410898813217995008957286, −2.96591719928827220916868482077, −2.54173841234764446716333668771, −1.44275253462130667219289765811,
1.44275253462130667219289765811, 2.54173841234764446716333668771, 2.96591719928827220916868482077, 3.87322410898813217995008957286, 4.34191286536892455889759876690, 5.73060171033419482492981785043, 6.49269437412168924571195937684, 7.41827881332084189538318303236, 7.948094450817424305881566572464, 9.205304305407575833315372102956
