| L(s) = 1 | + (17.5 − 10.1i)2-s + (49.6 + 86.0i)3-s + (−50.8 + 88.0i)4-s + 1.79e3i·5-s + (1.74e3 + 1.00e3i)6-s + (−1.67e3 − 967. i)7-s + 1.24e4i·8-s + (4.91e3 − 8.50e3i)9-s + (1.82e4 + 3.15e4i)10-s + (4.64e4 − 2.68e4i)11-s − 1.00e4·12-s + (1.53e3 + 1.02e5i)13-s − 3.92e4·14-s + (−1.54e5 + 8.93e4i)15-s + (9.98e4 + 1.72e5i)16-s + (1.98e5 − 3.43e5i)17-s + ⋯ |
| L(s) = 1 | + (0.775 − 0.447i)2-s + (0.353 + 0.613i)3-s + (−0.0992 + 0.171i)4-s + 1.28i·5-s + (0.548 + 0.316i)6-s + (−0.263 − 0.152i)7-s + 1.07i·8-s + (0.249 − 0.432i)9-s + (0.576 + 0.998i)10-s + (0.957 − 0.552i)11-s − 0.140·12-s + (0.0149 + 0.999i)13-s − 0.272·14-s + (−0.789 + 0.455i)15-s + (0.380 + 0.659i)16-s + (0.575 − 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.13576 + 1.23610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13576 + 1.23610i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.53e3 - 1.02e5i)T \) |
| good | 2 | \( 1 + (-17.5 + 10.1i)T + (256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-49.6 - 86.0i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 - 1.79e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 + (1.67e3 + 967. i)T + (2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-4.64e4 + 2.68e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (-1.98e5 + 3.43e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (4.35e5 + 2.51e5i)T + (1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (5.98e5 + 1.03e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-2.44e6 - 4.23e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 - 2.72e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + (-5.29e6 + 3.05e6i)T + (6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-2.28e7 + 1.31e7i)T + (1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-8.62e6 + 1.49e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 - 1.60e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 4.91e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-4.20e7 - 2.42e7i)T + (4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (6.58e7 - 1.13e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.08e8 + 6.27e7i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (2.59e8 + 1.50e8i)T + (2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 2.58e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 8.39e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.52e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + (9.87e8 - 5.70e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.20e9 - 6.95e8i)T + (3.80e17 + 6.58e17i)T^{2} \) |
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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
−18.06024108737122654753627674293, −16.36374141568508519001833085818, −14.59541987219427089310110095279, −14.06104264058391999902159098262, −12.15509077812315234236770069335, −10.81380272556161080379640685687, −9.089341505560391481275122465010, −6.71752363264388211156255735227, −4.13879548444323766298831068957, −2.94707842734536605108552284126,
1.24618016669424069828012787539, 4.35872015339458809919735330603, 6.01243077329158026047139753416, 8.028917077045440777336303530146, 9.771396616052600588332256376497, 12.52736044967245567091511105964, 13.09670915388072914942712160636, 14.53296616471286558187931271811, 15.88376795600529753958998267596, 17.27005423546639201265533939282
