| L(s) = 1 | + (−45.4 + 78.7i)2-s + (108. − 188. i)3-s + (−35.6 − 61.7i)4-s − 2.56e3·5-s + (9.88e3 + 1.71e4i)6-s + (7.73e4 + 1.33e5i)7-s − 7.38e5·8-s + (7.73e5 + 1.33e6i)9-s + (1.16e5 − 2.02e5i)10-s + (−3.24e6 + 5.62e6i)11-s − 1.55e4·12-s + (−7.09e6 − 1.58e7i)13-s − 1.40e7·14-s + (−2.79e5 + 4.83e5i)15-s + (3.38e7 − 5.86e7i)16-s + (−8.64e7 − 1.49e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.502 + 0.869i)2-s + (0.0861 − 0.149i)3-s + (−0.00435 − 0.00754i)4-s − 0.0735·5-s + (0.0865 + 0.149i)6-s + (0.248 + 0.430i)7-s − 0.995·8-s + (0.485 + 0.840i)9-s + (0.0369 − 0.0639i)10-s + (−0.552 + 0.956i)11-s − 0.00150·12-s + (−0.407 − 0.913i)13-s − 0.499·14-s + (−0.00633 + 0.0109i)15-s + (0.504 − 0.873i)16-s + (−0.868 − 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.175807 - 0.500788i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175807 - 0.500788i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (7.09e6 + 1.58e7i)T \) |
| good | 2 | \( 1 + (45.4 - 78.7i)T + (-4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-108. + 188. i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + 2.56e3T + 1.22e9T^{2} \) |
| 7 | \( 1 + (-7.73e4 - 1.33e5i)T + (-4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (3.24e6 - 5.62e6i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (8.64e7 + 1.49e8i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (4.18e7 + 7.25e7i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (4.20e8 - 7.28e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 + (-2.34e8 + 4.05e8i)T + (-5.13e18 - 8.88e18i)T^{2} \) |
| 31 | \( 1 - 1.39e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + (1.07e10 - 1.86e10i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (3.15e9 - 5.46e9i)T + (-4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (-4.87e9 - 8.44e9i)T + (-8.59e20 + 1.48e21i)T^{2} \) |
| 47 | \( 1 + 1.02e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.69e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + (-2.87e11 - 4.97e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-1.40e11 - 2.43e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-3.10e11 + 5.37e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (-3.15e11 - 5.46e11i)T + (-5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 - 8.55e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 6.94e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 1.69e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (2.51e12 - 4.35e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (-6.28e12 - 1.08e13i)T + (-3.36e25 + 5.82e25i)T^{2} \) |
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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
−17.53338081972128989424858923560, −15.92769394235946332968703845292, −15.24761611872492836108239532640, −13.33589158022848258720480794656, −11.80898439818609203384468407168, −9.805953361265565431591738964222, −8.098234790238763323995832813699, −7.12469220297110899034306265498, −5.12343928285480125029135395912, −2.43645212249725310389296951084,
0.23951139105313001804714072283, 1.90262815236319817049539134460, 3.86403912411669060604359335601, 6.33309795632848887097226404002, 8.572399746098276908103272528927, 10.04173038011483686246871383331, 11.15692521734662651259992953343, 12.56885061972581724606756158663, 14.40956608011824870261751253729, 15.78119450337164979373267800665
