| L(s) = 1 | + (−55.2 + 31.8i)2-s + (924. + 1.60e3i)3-s + (−2.06e3 + 3.56e3i)4-s − 5.77e4i·5-s + (−1.02e5 − 5.90e4i)6-s + (−1.48e5 − 8.58e4i)7-s − 7.85e5i·8-s + (−9.13e5 + 1.58e6i)9-s + (1.84e6 + 3.19e6i)10-s + (3.30e6 − 1.90e6i)11-s − 7.62e6·12-s + (−1.69e7 − 4.11e6i)13-s + 1.09e7·14-s + (9.25e7 − 5.34e7i)15-s + (8.17e6 + 1.41e7i)16-s + (6.56e7 − 1.13e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.610 + 0.352i)2-s + (0.732 + 1.26i)3-s + (−0.251 + 0.435i)4-s − 1.65i·5-s + (−0.894 − 0.516i)6-s + (−0.477 − 0.275i)7-s − 1.05i·8-s + (−0.572 + 0.992i)9-s + (0.582 + 1.00i)10-s + (0.562 − 0.324i)11-s − 0.737·12-s + (−0.971 − 0.236i)13-s + 0.388·14-s + (2.09 − 1.21i)15-s + (0.121 + 0.211i)16-s + (0.659 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.905665 - 0.369628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.905665 - 0.369628i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.69e7 + 4.11e6i)T \) |
| good | 2 | \( 1 + (55.2 - 31.8i)T + (4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-924. - 1.60e3i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + 5.77e4iT - 1.22e9T^{2} \) |
| 7 | \( 1 + (1.48e5 + 8.58e4i)T + (4.84e10 + 8.39e10i)T^{2} \) |
| 11 | \( 1 + (-3.30e6 + 1.90e6i)T + (1.72e13 - 2.98e13i)T^{2} \) |
| 17 | \( 1 + (-6.56e7 + 1.13e8i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-2.87e7 - 1.65e7i)T + (2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-2.37e8 - 4.12e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + (2.91e9 + 5.05e9i)T + (-5.13e18 + 8.88e18i)T^{2} \) |
| 31 | \( 1 - 1.32e9iT - 2.44e19T^{2} \) |
| 37 | \( 1 + (1.02e10 - 5.91e9i)T + (1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + (-3.16e10 + 1.82e10i)T + (4.62e20 - 8.01e20i)T^{2} \) |
| 43 | \( 1 + (-9.93e9 + 1.72e10i)T + (-8.59e20 - 1.48e21i)T^{2} \) |
| 47 | \( 1 + 3.84e10iT - 5.46e21T^{2} \) |
| 53 | \( 1 + 2.53e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + (5.56e11 + 3.21e11i)T + (5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (1.78e11 - 3.09e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-9.13e11 + 5.27e11i)T + (2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + (1.89e11 + 1.09e11i)T + (5.82e23 + 1.00e24i)T^{2} \) |
| 73 | \( 1 + 1.49e12iT - 1.67e24T^{2} \) |
| 79 | \( 1 - 3.07e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.85e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + (-4.51e12 + 2.60e12i)T + (1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + (-1.41e11 - 8.19e10i)T + (3.36e25 + 5.82e25i)T^{2} \) |
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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
−16.50448140101666146243169346406, −15.59317563633705905315802783468, −13.72754610992984762372070989945, −12.28868302899342238869782942039, −9.610626684692217166934735573671, −9.248299823007829303280350008288, −7.85891522825493052165765002044, −4.86854936519275054609025535099, −3.57432563272827654989692480869, −0.46311303271699605933677642615,
1.66825327583621587955389294815, 2.88444332647613789280859523634, 6.39340881832776086851432701371, 7.62735797829189538320920353746, 9.384306963339645196883106807202, 10.80344231754601364243323709855, 12.53911438496445077323955019943, 14.33228441853759568359303865701, 14.68998233577118180778268695909, 17.49422600916994920408138180979
