| L(s) = 1 | + (−27.7 + 16i)2-s + (−357. + 221. i)3-s + (511. − 886. i)4-s + (3.64e3 + 6.31e3i)5-s + (6.36e3 − 1.18e4i)6-s + (1.85e4 + 4.04e4i)7-s + 3.27e4i·8-s + (7.88e4 − 1.58e5i)9-s + (−2.02e5 − 1.16e5i)10-s + (4.93e5 + 2.84e5i)11-s + (1.34e4 + 4.30e5i)12-s − 1.29e6i·13-s + (−1.16e6 − 8.23e5i)14-s + (−2.70e6 − 1.45e6i)15-s + (−5.24e5 − 9.08e5i)16-s + (4.99e6 − 8.65e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.849 + 0.526i)3-s + (0.249 − 0.433i)4-s + (0.521 + 0.903i)5-s + (0.334 − 0.623i)6-s + (0.416 + 0.909i)7-s + 0.353i·8-s + (0.444 − 0.895i)9-s + (−0.638 − 0.368i)10-s + (0.923 + 0.533i)11-s + (0.0156 + 0.499i)12-s − 0.965i·13-s + (−0.576 − 0.409i)14-s + (−0.919 − 0.493i)15-s + (−0.125 − 0.216i)16-s + (0.853 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.25021 + 0.776349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25021 + 0.776349i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (27.7 - 16i)T \) |
| 3 | \( 1 + (357. - 221. i)T \) |
| 7 | \( 1 + (-1.85e4 - 4.04e4i)T \) |
| good | 5 | \( 1 + (-3.64e3 - 6.31e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-4.93e5 - 2.84e5i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.29e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (-4.99e6 + 8.65e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.48e6 + 4.89e6i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.68e7 + 9.73e6i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 5.92e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (-8.65e7 - 4.99e7i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (1.95e7 + 3.38e7i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 7.21e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.17e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.85e8 - 3.20e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-2.83e9 - 1.63e9i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-3.11e9 + 5.39e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (4.99e9 - 2.88e9i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.00e10 + 1.74e10i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 9.51e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (8.51e9 + 4.91e9i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (2.52e10 + 4.37e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 3.99e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-2.26e10 - 3.91e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.01e11iT - 7.15e21T^{2} \) |
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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
−14.23600768220544622670716158024, −12.18755632871708408293210467259, −11.24415988471447179988503476817, −10.05261688113711780073381711712, −9.203063732464622171051113947579, −7.31872064358848963420587498908, −6.11884501741636119533998756063, −5.04502675042804066155823809317, −2.79199266346683870304287789836, −0.885141710035693540701025928647,
1.04518928957524829391081674386, 1.46556623773048444426027257705, 4.09300031762235157036452410566, 5.70554399940598778794671277455, 7.10065673658898680215645085006, 8.452834416744723520917579331910, 9.824321474509699814901216478078, 11.04991254576356750736224932286, 12.03504065097999597230518987129, 13.12232599428925676289601782464
