| L(s) = 1 | − 2.24·2-s − 26.9·4-s − 53.3·5-s + 89.8·7-s + 132.·8-s + 120.·10-s − 225.·11-s + 725.·13-s − 202.·14-s + 564.·16-s − 44.9·17-s + 1.21e3·19-s + 1.43e3·20-s + 505.·22-s + 529·23-s − 274.·25-s − 1.63e3·26-s − 2.42e3·28-s − 2.59e3·29-s + 585.·31-s − 5.50e3·32-s + 101.·34-s − 4.79e3·35-s − 3.84e3·37-s − 2.72e3·38-s − 7.07e3·40-s + 4.29e3·41-s + ⋯ |
| L(s) = 1 | − 0.397·2-s − 0.842·4-s − 0.955·5-s + 0.693·7-s + 0.732·8-s + 0.379·10-s − 0.560·11-s + 1.19·13-s − 0.275·14-s + 0.551·16-s − 0.0377·17-s + 0.770·19-s + 0.804·20-s + 0.222·22-s + 0.208·23-s − 0.0878·25-s − 0.473·26-s − 0.583·28-s − 0.572·29-s + 0.109·31-s − 0.951·32-s + 0.0149·34-s − 0.662·35-s − 0.462·37-s − 0.306·38-s − 0.699·40-s + 0.399·41-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.24T + 32T^{2} \) |
| 5 | \( 1 + 53.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 89.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 225.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 725.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 44.9T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.21e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 2.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 585.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.84e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.29e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.95e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.17e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.42e4T + 8.58e9T^{2} \) |
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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
−11.04330000560153910776012132778, −10.03913933828197880498995860539, −8.799590328967264003117025322490, −8.153624802198670927122777441458, −7.32765958030257242449113122317, −5.57938446629014045389499577120, −4.48660328025438362697377426145, −3.45758554835424640939907484013, −1.34326012150385907222829818358, 0,
1.34326012150385907222829818358, 3.45758554835424640939907484013, 4.48660328025438362697377426145, 5.57938446629014045389499577120, 7.32765958030257242449113122317, 8.153624802198670927122777441458, 8.799590328967264003117025322490, 10.03913933828197880498995860539, 11.04330000560153910776012132778
