| L(s) = 1 | − 7.57·2-s + 25.3·4-s − 21.2·5-s − 80.9·7-s + 50.3·8-s + 160.·10-s + 209.·11-s − 933.·13-s + 613.·14-s − 1.19e3·16-s + 96.4·17-s + 1.14e3·19-s − 538.·20-s − 1.58e3·22-s + 529·23-s − 2.67e3·25-s + 7.06e3·26-s − 2.05e3·28-s − 4.17e3·29-s − 4.15e3·31-s + 7.42e3·32-s − 730.·34-s + 1.72e3·35-s − 8.19e3·37-s − 8.66e3·38-s − 1.07e3·40-s − 1.57e4·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.792·4-s − 0.380·5-s − 0.624·7-s + 0.278·8-s + 0.509·10-s + 0.522·11-s − 1.53·13-s + 0.836·14-s − 1.16·16-s + 0.0809·17-s + 0.727·19-s − 0.301·20-s − 0.699·22-s + 0.208·23-s − 0.855·25-s + 2.05·26-s − 0.494·28-s − 0.921·29-s − 0.776·31-s + 1.28·32-s − 0.108·34-s + 0.237·35-s − 0.984·37-s − 0.973·38-s − 0.105·40-s − 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4938411276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4938411276\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 7.57T + 32T^{2} \) |
| 5 | \( 1 + 21.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 80.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 209.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 933.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 96.4T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.14e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 4.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.19e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.57e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.70e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e4T + 8.58e9T^{2} \) |
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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
−11.37658876544178712667376137489, −10.13034300506150048369562527442, −9.598588158495133626377333266976, −8.686939991881019858282424650585, −7.51326592933823553903897609776, −6.94114826345322996502756744343, −5.26296224015247367358145497287, −3.74207013967762264047134403537, −2.09261237099294456414376959521, −0.51043309183873307051008279497,
0.51043309183873307051008279497, 2.09261237099294456414376959521, 3.74207013967762264047134403537, 5.26296224015247367358145497287, 6.94114826345322996502756744343, 7.51326592933823553903897609776, 8.686939991881019858282424650585, 9.598588158495133626377333266976, 10.13034300506150048369562527442, 11.37658876544178712667376137489
