| L(s) = 1 | + (−0.0710 − 0.265i)2-s + (−24.7 + 42.9i)3-s + (221. − 127. i)4-s + (−462. + 462. i)5-s + (13.1 + 3.52i)6-s + (−1.02e3 + 3.81e3i)7-s + (−99.4 − 99.4i)8-s + (2.05e3 + 3.55e3i)9-s + (155. + 89.8i)10-s + (3.12e3 − 837. i)11-s + 1.26e4i·12-s + (−2.79e4 + 5.81e3i)13-s + 1.08e3·14-s + (−8.39e3 − 3.13e4i)15-s + (3.27e4 − 5.67e4i)16-s + (1.30e5 − 7.55e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.00444 − 0.0165i)2-s + (−0.306 + 0.530i)3-s + (0.865 − 0.499i)4-s + (−0.740 + 0.740i)5-s + (0.0101 + 0.00272i)6-s + (−0.425 + 1.58i)7-s + (−0.0242 − 0.0242i)8-s + (0.312 + 0.541i)9-s + (0.0155 + 0.00898i)10-s + (0.213 − 0.0571i)11-s + 0.612i·12-s + (−0.979 + 0.203i)13-s + 0.0282·14-s + (−0.165 − 0.618i)15-s + (0.499 − 0.865i)16-s + (1.56 − 0.904i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.843526 + 0.997171i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.843526 + 0.997171i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.79e4 - 5.81e3i)T \) |
| good | 2 | \( 1 + (0.0710 + 0.265i)T + (-221. + 128i)T^{2} \) |
| 3 | \( 1 + (24.7 - 42.9i)T + (-3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (462. - 462. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + (1.02e3 - 3.81e3i)T + (-4.99e6 - 2.88e6i)T^{2} \) |
| 11 | \( 1 + (-3.12e3 + 837. i)T + (1.85e8 - 1.07e8i)T^{2} \) |
| 17 | \( 1 + (-1.30e5 + 7.55e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.29e5 + 3.46e4i)T + (1.47e10 + 8.49e9i)T^{2} \) |
| 23 | \( 1 + (-3.24e5 - 1.87e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (9.99e4 - 1.73e5i)T + (-2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-3.12e5 + 3.12e5i)T - 8.52e11iT^{2} \) |
| 37 | \( 1 + (-9.94e5 + 2.66e5i)T + (3.04e12 - 1.75e12i)T^{2} \) |
| 41 | \( 1 + (2.87e5 + 1.07e6i)T + (-6.91e12 + 3.99e12i)T^{2} \) |
| 43 | \( 1 + (4.76e5 - 2.75e5i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-5.38e6 - 5.38e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 - 8.68e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-5.31e5 + 1.98e6i)T + (-1.27e14 - 7.34e13i)T^{2} \) |
| 61 | \( 1 + (-2.34e6 - 4.06e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-5.59e6 - 2.08e7i)T + (-3.51e14 + 2.03e14i)T^{2} \) |
| 71 | \( 1 + (-1.49e7 - 3.99e6i)T + (5.59e14 + 3.22e14i)T^{2} \) |
| 73 | \( 1 + (8.84e6 + 8.84e6i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 3.39e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (3.90e6 - 3.90e6i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + (-1.54e7 + 4.12e6i)T + (3.40e15 - 1.96e15i)T^{2} \) |
| 97 | \( 1 + (1.30e8 + 3.49e7i)T + (6.78e15 + 3.91e15i)T^{2} \) |
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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
−18.98719289347309092344198069449, −16.59959845212871092875426664036, −15.47977192101957362993341200769, −14.80963950521565171403496676451, −12.17736434505811729755763601207, −11.14897300912684566693111714846, −9.673030675774199178454973276257, −7.26553221328853250416218691311, −5.46152740928593115935683110235, −2.70257185065756930349412077774,
0.851628188542775567370665600679, 3.87618323268135279852531421490, 6.76450458929514809782375197497, 7.86455010546835614231480297951, 10.38112719895393114231048164723, 12.12083560289946991769851258747, 12.83303071920105533525089019065, 14.99077174038696787587420669487, 16.73922474765730425818848703040, 17.00099649364695226866137659087
