| L(s) = 1 | + 1.73·3-s + (4.59 − 1.96i)5-s + (−6.81 − 1.57i)7-s + 2.99·9-s + 8.15·11-s + 14.6·13-s + (7.96 − 3.40i)15-s + 5.81·17-s + 33.7i·19-s + (−11.8 − 2.73i)21-s + 37.2i·23-s + (17.2 − 18.0i)25-s + 5.19·27-s − 9.25·29-s − 19.2i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (0.919 − 0.393i)5-s + (−0.974 − 0.225i)7-s + 0.333·9-s + 0.741·11-s + 1.12·13-s + (0.530 − 0.226i)15-s + 0.342·17-s + 1.77i·19-s + (−0.562 − 0.130i)21-s + 1.61i·23-s + (0.691 − 0.722i)25-s + 0.192·27-s − 0.319·29-s − 0.621i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.072127281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072127281\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + (-4.59 + 1.96i)T \) |
| 7 | \( 1 + (6.81 + 1.57i)T \) |
| good | 11 | \( 1 - 8.15T + 121T^{2} \) |
| 13 | \( 1 - 14.6T + 169T^{2} \) |
| 17 | \( 1 - 5.81T + 289T^{2} \) |
| 19 | \( 1 - 33.7iT - 361T^{2} \) |
| 23 | \( 1 - 37.2iT - 529T^{2} \) |
| 29 | \( 1 + 9.25T + 841T^{2} \) |
| 31 | \( 1 + 19.2iT - 961T^{2} \) |
| 37 | \( 1 - 63.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 23.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 71.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 42.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.99iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 124.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 16.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 82.7T + 9.40e3T^{2} \) |
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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
−9.326328660889488784843118289531, −8.495313568291023647947365929993, −7.73624147872741421164279287090, −6.58453384230508389914247473686, −6.09254449323606894673212405650, −5.22781406851396509817882087152, −3.74482086376048347242277377206, −3.47763209806139290726974904765, −1.95647680729935982419902387024, −1.13005554872866252270083971073,
0.890825454789922708726662814747, 2.25426204183269635688512964842, 3.01561300630315154652696111294, 3.90621356717449811370220062768, 5.06908076240157489969951688341, 6.29888445794112050463904335576, 6.47883531275213618683537497231, 7.44244271535144200497986314493, 8.750346112504834274786153919028, 9.066023083358732649377090055019
