| L(s) = 1 | + (410. − 711. i)3-s + (456. + 790. i)5-s + (−4.18e4 + 1.50e4i)7-s + (−2.48e5 − 4.30e5i)9-s + (5.68e4 − 9.84e4i)11-s − 1.60e6·13-s + 7.49e5·15-s + (4.03e5 − 6.98e5i)17-s + (7.33e6 + 1.27e7i)19-s + (−6.50e6 + 3.59e7i)21-s + (−2.21e7 − 3.83e7i)23-s + (2.39e7 − 4.15e7i)25-s − 2.62e8·27-s − 5.24e7·29-s + (−6.88e7 + 1.19e8i)31-s + ⋯ |
| L(s) = 1 | + (0.975 − 1.68i)3-s + (0.0652 + 0.113i)5-s + (−0.941 + 0.337i)7-s + (−1.40 − 2.43i)9-s + (0.106 − 0.184i)11-s − 1.19·13-s + 0.254·15-s + (0.0689 − 0.119i)17-s + (0.679 + 1.17i)19-s + (−0.347 + 1.92i)21-s + (−0.717 − 1.24i)23-s + (0.491 − 0.851i)25-s − 3.52·27-s − 0.475·29-s + (−0.431 + 0.748i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.156034 + 1.28083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.156034 + 1.28083i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (4.18e4 - 1.50e4i)T \) |
| good | 3 | \( 1 + (-410. + 711. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + (-456. - 790. i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-5.68e4 + 9.84e4i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.60e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-4.03e5 + 6.98e5i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-7.33e6 - 1.27e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (2.21e7 + 3.83e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 5.24e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (6.88e7 - 1.19e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-4.31e7 - 7.48e7i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 8.15e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.31e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.44e9 + 2.49e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.70e9 - 2.95e9i)T + (-4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-2.91e9 + 5.04e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.01e8 + 3.49e8i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (5.18e8 - 8.98e8i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.07e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-1.04e10 + 1.80e10i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.94e10 + 3.37e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 3.42e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (3.37e10 + 5.83e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 7.18e10T + 7.15e21T^{2} \) |
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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
−13.97807253017264959252116533095, −12.65001730861556571726544029784, −12.14076427075964573142655943616, −9.739952595301721005495360165212, −8.428708834592030129388360683277, −7.21592366292038543184300861817, −6.13155640605559221696832989988, −3.20057267362891665967134621883, −2.07549257676684420254683039013, −0.37182510236817209686197441861,
2.68331895555402037699931674557, 3.86773608933873252768033637031, 5.21407237448358239253286661731, 7.58035941977276708464989607792, 9.383309196749757882745406011446, 9.696948872729818994043131567291, 11.15115656895917587165587298764, 13.16995777262487158769347769270, 14.35186587108501004349929738339, 15.36900809921201538847017934655
