| L(s) = 1 | − 6.53·2-s + 9·3-s + 10.6·4-s − 25·5-s − 58.8·6-s + 229.·7-s + 139.·8-s + 81·9-s + 163.·10-s + 284.·11-s + 96.2·12-s − 169·13-s − 1.49e3·14-s − 225·15-s − 1.25e3·16-s + 1.58e3·17-s − 529.·18-s − 2.17e3·19-s − 267.·20-s + 2.06e3·21-s − 1.85e3·22-s + 1.12e3·23-s + 1.25e3·24-s + 625·25-s + 1.10e3·26-s + 729·27-s + 2.44e3·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.577·3-s + 0.334·4-s − 0.447·5-s − 0.666·6-s + 1.76·7-s + 0.769·8-s + 0.333·9-s + 0.516·10-s + 0.708·11-s + 0.192·12-s − 0.277·13-s − 2.04·14-s − 0.258·15-s − 1.22·16-s + 1.32·17-s − 0.385·18-s − 1.38·19-s − 0.149·20-s + 1.01·21-s − 0.818·22-s + 0.444·23-s + 0.444·24-s + 0.200·25-s + 0.320·26-s + 0.192·27-s + 0.590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.522192229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522192229\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 6.53T + 32T^{2} \) |
| 7 | \( 1 - 229.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 284.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.09e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.37e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.70e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.28e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.76e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.57e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.66e5T + 8.58e9T^{2} \) |
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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
−11.32952578594256432972320943534, −10.53985567168061452565547128906, −9.437630493872892147972184468424, −8.398799374891618187844901138770, −8.039365078772438658627480418585, −7.01848904783579682512590348488, −5.03254307490986481327045238858, −3.98892755298893769311121505235, −2.00060033565779870236323301538, −0.957867069670270083209848156613,
0.957867069670270083209848156613, 2.00060033565779870236323301538, 3.98892755298893769311121505235, 5.03254307490986481327045238858, 7.01848904783579682512590348488, 8.039365078772438658627480418585, 8.398799374891618187844901138770, 9.437630493872892147972184468424, 10.53985567168061452565547128906, 11.32952578594256432972320943534
