| L(s) = 1 | + (−11.1 + 19.2i)2-s + (128. − 222. i)3-s + (8.30 + 14.3i)4-s − 2.48e3·5-s + (2.85e3 + 4.94e3i)6-s + (−83.9 − 145. i)7-s − 1.17e4·8-s + (−2.30e4 − 3.99e4i)9-s + (2.76e4 − 4.79e4i)10-s + (−8.51e3 + 1.47e4i)11-s + 4.25e3·12-s + (−5.25e4 − 8.85e4i)13-s + 3.73e3·14-s + (−3.19e5 + 5.52e5i)15-s + (1.26e5 − 2.19e5i)16-s + (4.59e4 + 7.96e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.491 + 0.851i)2-s + (0.914 − 1.58i)3-s + (0.0162 + 0.0280i)4-s − 1.78·5-s + (0.899 + 1.55i)6-s + (−0.0132 − 0.0228i)7-s − 1.01·8-s + (−1.17 − 2.02i)9-s + (0.875 − 1.51i)10-s + (−0.175 + 0.303i)11-s + 0.0593·12-s + (−0.510 − 0.860i)13-s + 0.0259·14-s + (−1.62 + 2.81i)15-s + (0.483 − 0.837i)16-s + (0.133 + 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.199795 - 0.478615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.199795 - 0.478615i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (5.25e4 + 8.85e4i)T \) |
| good | 2 | \( 1 + (11.1 - 19.2i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-128. + 222. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + 2.48e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + (83.9 + 145. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (8.51e3 - 1.47e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (-4.59e4 - 7.96e4i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.47e5 + 2.54e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-4.52e4 + 7.84e4i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.02e6 - 1.77e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 - 6.02e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (4.13e6 - 7.16e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-9.85e6 + 1.70e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (4.32e6 + 7.48e6i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + 3.97e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.94e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (1.97e7 + 3.41e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (4.17e7 + 7.23e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (6.92e7 - 1.20e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (8.12e5 + 1.40e6i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 3.56e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.35e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-3.60e8 + 6.24e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (6.31e8 + 1.09e9i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−17.37730048425409160902876498972, −15.61101007198078823991461227573, −14.75034566394070683414712821057, −12.76319824483730991984224439980, −11.88029752699943732264937784687, −8.530314718556410745013469945632, −7.77381266611212831972133837346, −6.91155708533974458389690854321, −3.08677220731179547248248081493, −0.29055972023305159191947684288,
3.01332463398568014410703597748, 4.33903618741789061525753147965, 8.160151751666816530599537521117, 9.429731853708401615072451919649, 10.78196674276022864772693632620, 11.81346674348086979453835902194, 14.54859931250890709614324196254, 15.45408854007354843002044691427, 16.37866606451237047746818495197, 19.05616091593180221670297731691
