| L(s) = 1 | + (−39.1 − 22.6i)2-s + (1.02e3 + 1.77e3i)4-s + (1.25e4 − 7.23e3i)5-s + (−4.86e4 + 8.42e4i)7-s − 9.26e4i·8-s − 6.54e5·10-s + (1.20e6 + 6.96e5i)11-s + (3.01e6 + 5.21e6i)13-s + (3.81e6 − 2.20e6i)14-s + (−2.09e6 + 3.63e6i)16-s + 2.96e7i·17-s − 4.55e7·19-s + (2.56e7 + 1.48e7i)20-s + (−3.15e7 − 5.46e7i)22-s + (8.46e7 − 4.88e7i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.801 − 0.462i)5-s + (−0.413 + 0.715i)7-s − 0.353i·8-s − 0.654·10-s + (0.681 + 0.393i)11-s + (0.623 + 1.08i)13-s + (0.506 − 0.292i)14-s + (−0.125 + 0.216i)16-s + 1.22i·17-s − 0.967·19-s + (0.400 + 0.231i)20-s + (−0.278 − 0.481i)22-s + (0.571 − 0.330i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.9647544646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9647544646\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (39.1 + 22.6i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.25e4 + 7.23e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 7 | \( 1 + (4.86e4 - 8.42e4i)T + (-6.92e9 - 1.19e10i)T^{2} \) |
| 11 | \( 1 + (-1.20e6 - 6.96e5i)T + (1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 + (-3.01e6 - 5.21e6i)T + (-1.16e13 + 2.01e13i)T^{2} \) |
| 17 | \( 1 - 2.96e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 4.55e7T + 2.21e15T^{2} \) |
| 23 | \( 1 + (-8.46e7 + 4.88e7i)T + (1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 + (-4.17e8 - 2.41e8i)T + (1.76e17 + 3.06e17i)T^{2} \) |
| 31 | \( 1 + (-2.34e8 - 4.06e8i)T + (-3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + 4.39e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + (4.92e9 - 2.84e9i)T + (1.12e19 - 1.95e19i)T^{2} \) |
| 43 | \( 1 + (2.15e9 - 3.72e9i)T + (-1.99e19 - 3.46e19i)T^{2} \) |
| 47 | \( 1 + (-3.32e9 - 1.91e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + 2.52e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 + (5.60e10 - 3.23e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-1.61e10 + 2.79e10i)T + (-1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (2.90e10 + 5.03e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 + 3.98e10iT - 1.64e22T^{2} \) |
| 73 | \( 1 - 1.63e11T + 2.29e22T^{2} \) |
| 79 | \( 1 + (-9.35e10 + 1.62e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 + (6.69e10 + 3.86e10i)T + (5.34e22 + 9.25e22i)T^{2} \) |
| 89 | \( 1 + 9.19e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + (-7.16e11 + 1.24e12i)T + (-3.46e23 - 6.00e23i)T^{2} \) |
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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
−10.87463732106686767240058721273, −9.901599300935314028229339591242, −8.950769116287194635634093083791, −8.541096821434759781319291831351, −6.75979560799613713427083578252, −6.11359787598469067769192189651, −4.65783062130313407035756058763, −3.37737488176548147254699184529, −1.92344714285554323284481939511, −1.44358407968239875344473807128,
0.23317774638859020419994233908, 1.14926742678328889574990515537, 2.50942246125231692004158020395, 3.67499243924565229537210171579, 5.28850551961949450340083965675, 6.36284907667491134307718263950, 7.01740652962058654189431370924, 8.285763309492110690740681437700, 9.286530024212638608488419501346, 10.25974123933855002443169971531
