| L(s) = 1 | + (43.8 − 76.0i)2-s + (−272. + 471. i)3-s + (−2.82e3 − 4.90e3i)4-s − 4.93e3·5-s + (2.39e4 + 4.14e4i)6-s + (2.47e4 + 4.28e4i)7-s − 3.16e5·8-s + (−5.97e4 − 1.03e5i)9-s + (−2.16e5 + 3.75e5i)10-s + (−2.42e5 + 4.19e5i)11-s + 3.08e6·12-s + (−1.30e6 − 2.90e5i)13-s + 4.33e6·14-s + (1.34e6 − 2.32e6i)15-s + (−8.11e6 + 1.40e7i)16-s + (−1.75e6 − 3.03e6i)17-s + ⋯ |
| L(s) = 1 | + (0.969 − 1.67i)2-s + (−0.646 + 1.12i)3-s + (−1.38 − 2.39i)4-s − 0.706·5-s + (1.25 + 2.17i)6-s + (0.555 + 0.962i)7-s − 3.41·8-s + (−0.337 − 0.583i)9-s + (−0.685 + 1.18i)10-s + (−0.453 + 0.785i)11-s + 3.57·12-s + (−0.976 − 0.216i)13-s + 2.15·14-s + (0.457 − 0.791i)15-s + (−1.93 + 3.35i)16-s + (−0.299 − 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0367 - 0.999i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.0367 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.134209 + 0.139234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134209 + 0.139234i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.30e6 + 2.90e5i)T \) |
| good | 2 | \( 1 + (-43.8 + 76.0i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (272. - 471. i)T + (-8.85e4 - 1.53e5i)T^{2} \) |
| 5 | \( 1 + 4.93e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + (-2.47e4 - 4.28e4i)T + (-9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (2.42e5 - 4.19e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 17 | \( 1 + (1.75e6 + 3.03e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (5.89e6 + 1.02e7i)T + (-5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.38e7 + 2.39e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + (7.11e7 - 1.23e8i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 - 9.57e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-1.86e7 + 3.23e7i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (-2.35e8 + 4.07e8i)T + (-2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-7.60e8 - 1.31e9i)T + (-4.64e17 + 8.04e17i)T^{2} \) |
| 47 | \( 1 + 3.60e6T + 2.47e18T^{2} \) |
| 53 | \( 1 + 7.15e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + (1.87e9 + 3.24e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.41e9 - 2.44e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-5.15e8 + 8.92e8i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + (3.10e9 + 5.37e9i)T + (-1.15e20 + 2.00e20i)T^{2} \) |
| 73 | \( 1 + 1.47e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.97e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.26e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (4.33e10 - 7.50e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + (-3.43e10 - 5.95e10i)T + (-3.57e21 + 6.19e21i)T^{2} \) |
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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
−17.86992192831139905527524075124, −15.51174090481969395650322576760, −14.76664080763251662149861701164, −12.68175410571231033089293471358, −11.63159684918091514463698992107, −10.69461619740261290681675185656, −9.365013842471059972076540692020, −5.18227082733310751192628779988, −4.45020352514597040014635022540, −2.50588082814035406627580664561,
0.06977327589187610660690133223, 4.12531741931599074422257630956, 5.85156474233266166360122975445, 7.27042937130966414727072976859, 7.991034561392413482491455945181, 11.70341736054660961847647365337, 12.93587371395753211961261378022, 13.99785904922227388573561703579, 15.36505338534279077509294875979, 16.86884805181472217374879359008
