| L(s) = 1 | + 10.6·2-s + 19.4·3-s + 81.9·4-s + 25·5-s + 207.·6-s − 110.·7-s + 533.·8-s + 133.·9-s + 266.·10-s + 121·11-s + 1.59e3·12-s + 401.·13-s − 1.18e3·14-s + 485.·15-s + 3.07e3·16-s − 358.·17-s + 1.43e3·18-s + 361·19-s + 2.04e3·20-s − 2.15e3·21-s + 1.29e3·22-s + 165.·23-s + 1.03e4·24-s + 625·25-s + 4.28e3·26-s − 2.11e3·27-s − 9.08e3·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 1.24·3-s + 2.56·4-s + 0.447·5-s + 2.35·6-s − 0.855·7-s + 2.94·8-s + 0.551·9-s + 0.843·10-s + 0.301·11-s + 3.19·12-s + 0.658·13-s − 1.61·14-s + 0.557·15-s + 2.99·16-s − 0.300·17-s + 1.04·18-s + 0.229·19-s + 1.14·20-s − 1.06·21-s + 0.568·22-s + 0.0654·23-s + 3.66·24-s + 0.200·25-s + 1.24·26-s − 0.558·27-s − 2.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(14.62526066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.62526066\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 10.6T + 32T^{2} \) |
| 3 | \( 1 - 19.4T + 243T^{2} \) |
| 7 | \( 1 + 110.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 401.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 358.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 165.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.02e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 718.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.24e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.73e4T + 8.58e9T^{2} \) |
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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
−9.184346701712640128207931677572, −8.234998502040975221635048724355, −7.21883824149932242360466149201, −6.37419029805625778673322567782, −5.83558257162710544620927186407, −4.59024430129406324589997963327, −3.83539615547315858531377809867, −2.93631250205747794417615448682, −2.54593523222106176333479508544, −1.29580891629685344287953899874,
1.29580891629685344287953899874, 2.54593523222106176333479508544, 2.93631250205747794417615448682, 3.83539615547315858531377809867, 4.59024430129406324589997963327, 5.83558257162710544620927186407, 6.37419029805625778673322567782, 7.21883824149932242360466149201, 8.234998502040975221635048724355, 9.184346701712640128207931677572
