| L(s) = 1 | + (−4.98 + 18.6i)2-s + (51.4 + 89.0i)3-s + (−100. − 57.8i)4-s + (663. + 663. i)5-s + (−1.91e3 + 512. i)6-s + (−974. − 3.63e3i)7-s + (−1.91e3 + 1.91e3i)8-s + (−2.00e3 + 3.47e3i)9-s + (−1.56e4 + 9.04e3i)10-s + (2.95e3 + 790. i)11-s − 1.18e4i·12-s + (1.33e4 − 2.52e4i)13-s + 7.25e4·14-s + (−2.49e4 + 9.31e4i)15-s + (−4.08e4 − 7.08e4i)16-s + (7.61e4 + 4.39e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.311 + 1.16i)2-s + (0.634 + 1.09i)3-s + (−0.391 − 0.225i)4-s + (1.06 + 1.06i)5-s + (−1.47 + 0.395i)6-s + (−0.405 − 1.51i)7-s + (−0.467 + 0.467i)8-s + (−0.305 + 0.529i)9-s + (−1.56 + 0.904i)10-s + (0.201 + 0.0540i)11-s − 0.573i·12-s + (0.466 − 0.884i)13-s + 1.88·14-s + (−0.493 + 1.84i)15-s + (−0.623 − 1.08i)16-s + (0.911 + 0.526i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.405749 + 1.78439i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.405749 + 1.78439i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.33e4 + 2.52e4i)T \) |
| good | 2 | \( 1 + (4.98 - 18.6i)T + (-221. - 128i)T^{2} \) |
| 3 | \( 1 + (-51.4 - 89.0i)T + (-3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-663. - 663. i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + (974. + 3.63e3i)T + (-4.99e6 + 2.88e6i)T^{2} \) |
| 11 | \( 1 + (-2.95e3 - 790. i)T + (1.85e8 + 1.07e8i)T^{2} \) |
| 17 | \( 1 + (-7.61e4 - 4.39e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-4.81e4 + 1.28e4i)T + (1.47e10 - 8.49e9i)T^{2} \) |
| 23 | \( 1 + (3.98e5 - 2.30e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (3.95e4 + 6.84e4i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (2.47e5 + 2.47e5i)T + 8.52e11iT^{2} \) |
| 37 | \( 1 + (-6.99e5 - 1.87e5i)T + (3.04e12 + 1.75e12i)T^{2} \) |
| 41 | \( 1 + (-2.90e5 + 1.08e6i)T + (-6.91e12 - 3.99e12i)T^{2} \) |
| 43 | \( 1 + (-1.61e6 - 9.31e5i)T + (5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-1.25e6 + 1.25e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + 2.09e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (5.48e6 + 2.04e7i)T + (-1.27e14 + 7.34e13i)T^{2} \) |
| 61 | \( 1 + (1.17e7 - 2.03e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-4.35e6 + 1.62e7i)T + (-3.51e14 - 2.03e14i)T^{2} \) |
| 71 | \( 1 + (3.29e7 - 8.84e6i)T + (5.59e14 - 3.22e14i)T^{2} \) |
| 73 | \( 1 + (-3.93e7 + 3.93e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 2.79e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + (3.81e7 + 3.81e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + (-3.13e6 - 8.39e5i)T + (3.40e15 + 1.96e15i)T^{2} \) |
| 97 | \( 1 + (-3.75e7 + 1.00e7i)T + (6.78e15 - 3.91e15i)T^{2} \) |
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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
−18.03579543127729922193464646793, −16.99169029189712494823293361980, −15.80566259842052471393799756004, −14.60032748979975856824862818392, −13.79793534342956295442038881660, −10.51864151808739571220424449915, −9.656967583684157562296118390140, −7.63837011019002230795970574916, −6.08641485775151085573299915746, −3.42568968085177463382927396258,
1.39564105147261135708871742061, 2.45261495767165400425564426670, 6.05283938224126978962503564224, 8.703312714121152957258584199345, 9.604664981104903243062671250335, 12.03499962067570814226202225971, 12.69584673617265234899530183658, 14.00367899407183365590608982673, 16.20075227382587834414332888586, 18.21534494789651734325580600652
