| L(s) = 1 | + (−103. + 60.0i)2-s + (−41.2 − 71.4i)3-s + (3.10e3 − 5.38e3i)4-s + 2.80e4i·5-s + (8.58e3 + 4.95e3i)6-s + (−3.89e5 − 2.25e5i)7-s − 2.36e5i·8-s + (7.93e5 − 1.37e6i)9-s + (−1.68e6 − 2.91e6i)10-s + (−1.14e6 + 6.59e5i)11-s − 5.13e5·12-s + (−9.19e6 + 1.47e7i)13-s + 5.40e7·14-s + (2.00e6 − 1.15e6i)15-s + (3.96e7 + 6.87e7i)16-s + (4.23e7 − 7.33e7i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 + 0.663i)2-s + (−0.0326 − 0.0566i)3-s + (0.379 − 0.657i)4-s + 0.802i·5-s + (0.0750 + 0.0433i)6-s + (−1.25 − 0.723i)7-s − 0.319i·8-s + (0.497 − 0.862i)9-s + (−0.532 − 0.922i)10-s + (−0.194 + 0.112i)11-s − 0.0496·12-s + (−0.528 + 0.848i)13-s + 1.91·14-s + (0.0454 − 0.0262i)15-s + (0.591 + 1.02i)16-s + (0.425 − 0.736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.710883 + 0.173036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.710883 + 0.173036i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (9.19e6 - 1.47e7i)T \) |
| good | 2 | \( 1 + (103. - 60.0i)T + (4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (41.2 + 71.4i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 - 2.80e4iT - 1.22e9T^{2} \) |
| 7 | \( 1 + (3.89e5 + 2.25e5i)T + (4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (1.14e6 - 6.59e5i)T + (1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (-4.23e7 + 7.33e7i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-2.97e8 - 1.72e8i)T + (2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (2.75e8 + 4.76e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + (-2.33e9 - 4.04e9i)T + (-5.13e18 + 8.88e18i)T^{2} \) |
| 31 | \( 1 + 1.03e9iT - 2.44e19T^{2} \) |
| 37 | \( 1 + (-5.72e9 + 3.30e9i)T + (1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (6.36e9 - 3.67e9i)T + (4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (-2.36e10 + 4.09e10i)T + (-8.59e20 - 1.48e21i)T^{2} \) |
| 47 | \( 1 - 1.79e10iT - 5.46e21T^{2} \) |
| 53 | \( 1 - 2.47e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + (4.21e11 + 2.43e11i)T + (5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-3.09e11 + 5.36e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-2.61e11 + 1.50e11i)T + (2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (-1.01e12 - 5.86e11i)T + (5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 - 3.22e11iT - 1.67e24T^{2} \) |
| 79 | \( 1 - 2.17e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.45e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + (-1.56e12 + 9.05e11i)T + (1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (-8.29e12 - 4.78e12i)T + (3.36e25 + 5.82e25i)T^{2} \) |
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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
−16.67817611044245341915964724407, −15.85696828757588481448477228950, −14.22373714595091515312189162481, −12.40646006380367934921188812622, −10.20736680447163468365802302814, −9.428636397997600990893709411923, −7.30352780958278873373484953064, −6.61124524374155665682837541754, −3.46955126323951228935373301558, −0.67913992438794199280303814753,
0.854147546392410486555744297318, 2.68565271935278808403677685351, 5.37287423821611883577211311816, 7.891393660036914260836553482194, 9.327072768103308650498098164320, 10.25518030296161135035303640933, 12.03536357607098248173070823388, 13.27384978807163463868247198533, 15.64954756802078515801539511045, 16.70868893547744729223241884310
