| L(s) = 1 | + (22.6 − 39.1i)2-s + (985. − 569. i)3-s + (−1.02e3 − 1.77e3i)4-s + (−1.21e4 − 7.00e3i)5-s − 5.15e4i·6-s + (−1.03e5 − 5.52e4i)7-s − 9.26e4·8-s + (3.81e5 − 6.61e5i)9-s + (−5.49e5 + 3.17e5i)10-s + (1.52e6 + 2.63e6i)11-s + (−2.01e6 − 1.16e6i)12-s − 8.06e6i·13-s + (−4.51e6 + 2.82e6i)14-s − 1.59e7·15-s + (−2.09e6 + 3.63e6i)16-s + (−2.92e6 + 1.68e6i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (1.35 − 0.780i)3-s + (−0.249 − 0.433i)4-s + (−0.776 − 0.448i)5-s − 1.10i·6-s + (−0.882 − 0.469i)7-s − 0.353·8-s + (0.718 − 1.24i)9-s + (−0.549 + 0.317i)10-s + (0.858 + 1.48i)11-s + (−0.676 − 0.390i)12-s − 1.67i·13-s + (−0.599 + 0.374i)14-s − 1.39·15-s + (−0.125 + 0.216i)16-s + (−0.121 + 0.0699i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.520114 - 2.37332i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520114 - 2.37332i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-22.6 + 39.1i)T \) |
| 7 | \( 1 + (1.03e5 + 5.52e4i)T \) |
| good | 3 | \( 1 + (-985. + 569. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (1.21e4 + 7.00e3i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-1.52e6 - 2.63e6i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 + 8.06e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (2.92e6 - 1.68e6i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (1.38e7 + 7.97e6i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (-6.09e7 + 1.05e8i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 - 1.10e9T + 3.53e17T^{2} \) |
| 31 | \( 1 + (1.49e8 - 8.60e7i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-8.32e8 + 1.44e9i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 + 2.84e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 3.52e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (4.75e9 + 2.74e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-3.54e9 - 6.14e9i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (-2.94e10 + 1.69e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-6.13e10 - 3.53e10i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (-5.76e10 - 9.97e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 + 5.03e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (6.57e10 - 3.79e10i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (1.28e11 - 2.22e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 - 2.08e9iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (7.55e11 + 4.36e11i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 + 8.85e11iT - 6.93e23T^{2} \) |
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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
−15.56425702285329950138264812557, −14.43012406527498935156702292248, −12.94773676554871332448915843224, −12.39817895246109434078296319186, −10.03793838437048300510297719704, −8.505613277918970761101196613632, −7.04915916453523053916185836525, −4.09889098999182408749485717490, −2.68274583975405758395560404487, −0.830926302728564102996696584925,
3.07342656713621146837325400337, 4.03180123846485385662332269098, 6.58892215230214312838510428431, 8.440057933685025626374438207878, 9.392357726449818778390058241858, 11.59826560834630161190529346489, 13.66286806304458671681432136058, 14.54965558986441168899499142460, 15.70615910191809071170416920980, 16.48053339345864909091480695106
