| L(s) = 1 | + 2.52·2-s − 25.6·4-s + 62.7·5-s + 238.·7-s − 145.·8-s + 158.·10-s + 620.·11-s + 768.·13-s + 602.·14-s + 451.·16-s − 632.·17-s − 910.·19-s − 1.60e3·20-s + 1.56e3·22-s − 1.01e3·23-s + 813.·25-s + 1.94e3·26-s − 6.10e3·28-s − 3.47e3·29-s + 890.·31-s + 5.80e3·32-s − 1.59e3·34-s + 1.49e4·35-s + 4.04e3·37-s − 2.30e3·38-s − 9.13e3·40-s − 1.68e3·41-s + ⋯ |
| L(s) = 1 | + 0.446·2-s − 0.800·4-s + 1.12·5-s + 1.83·7-s − 0.804·8-s + 0.501·10-s + 1.54·11-s + 1.26·13-s + 0.821·14-s + 0.441·16-s − 0.530·17-s − 0.578·19-s − 0.898·20-s + 0.691·22-s − 0.398·23-s + 0.260·25-s + 0.563·26-s − 1.47·28-s − 0.767·29-s + 0.166·31-s + 1.00·32-s − 0.236·34-s + 2.06·35-s + 0.485·37-s − 0.258·38-s − 0.902·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.825456720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.825456720\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 + 1.68e3T \) |
| good | 2 | \( 1 - 2.52T + 32T^{2} \) |
| 5 | \( 1 - 62.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 238.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 620.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 768.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 632.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 910.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.01e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 890.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.04e3T + 6.93e7T^{2} \) |
| 43 | \( 1 - 1.32e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.44e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.67e5T + 8.58e9T^{2} \) |
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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
−10.73391887670836443384578793882, −9.411419246837418276118225729034, −8.857711934931610870353016714544, −8.008073139887495113793996875771, −6.38846644966914789669329507897, −5.68289220585623326253555439261, −4.59917646489817114617349175510, −3.85398744474090261629092751970, −1.96422690973305153187456750770, −1.12563606147641807703364289517,
1.12563606147641807703364289517, 1.96422690973305153187456750770, 3.85398744474090261629092751970, 4.59917646489817114617349175510, 5.68289220585623326253555439261, 6.38846644966914789669329507897, 8.008073139887495113793996875771, 8.857711934931610870353016714544, 9.411419246837418276118225729034, 10.73391887670836443384578793882
