| L(s) = 1 | + (−15.0 − 5.47i)2-s + (−40.3 − 228. i)3-s + (196. + 164. i)4-s + (931. − 781. i)5-s + (−645. + 3.66e3i)6-s + (−2.74e3 + 4.76e3i)7-s + (−2.04e3 − 3.54e3i)8-s + (−3.22e4 + 1.17e4i)9-s + (−1.82e4 + 6.65e3i)10-s + (−3.85e4 − 6.67e4i)11-s + (2.97e4 − 5.15e4i)12-s + (−2.99e4 + 1.70e5i)13-s + (6.73e4 − 5.65e4i)14-s + (−2.16e5 − 1.81e5i)15-s + (1.13e4 + 6.45e4i)16-s + (1.07e5 + 3.92e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.287 − 1.63i)3-s + (0.383 + 0.321i)4-s + (0.666 − 0.559i)5-s + (−0.203 + 1.15i)6-s + (−0.432 + 0.749i)7-s + (−0.176 − 0.306i)8-s + (−1.63 + 0.596i)9-s + (−0.578 + 0.210i)10-s + (−0.794 − 1.37i)11-s + (0.414 − 0.717i)12-s + (−0.291 + 1.65i)13-s + (0.468 − 0.393i)14-s + (−1.10 − 0.926i)15-s + (0.0434 + 0.246i)16-s + (0.312 + 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.105297 + 0.0691997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105297 + 0.0691997i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.0 + 5.47i)T \) |
| 19 | \( 1 + (5.67e5 + 1.13e4i)T \) |
| good | 3 | \( 1 + (40.3 + 228. i)T + (-1.84e4 + 6.73e3i)T^{2} \) |
| 5 | \( 1 + (-931. + 781. i)T + (3.39e5 - 1.92e6i)T^{2} \) |
| 7 | \( 1 + (2.74e3 - 4.76e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (3.85e4 + 6.67e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (2.99e4 - 1.70e5i)T + (-9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (-1.07e5 - 3.92e4i)T + (9.08e10 + 7.62e10i)T^{2} \) |
| 23 | \( 1 + (-1.35e5 - 1.13e5i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-2.87e6 + 1.04e6i)T + (1.11e13 - 9.32e12i)T^{2} \) |
| 31 | \( 1 + (4.02e6 - 6.97e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 1.62e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.56e6 + 1.45e7i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-1.89e7 + 1.58e7i)T + (8.72e13 - 4.94e14i)T^{2} \) |
| 47 | \( 1 + (2.84e7 - 1.03e7i)T + (8.57e14 - 7.19e14i)T^{2} \) |
| 53 | \( 1 + (6.65e7 + 5.58e7i)T + (5.72e14 + 3.24e15i)T^{2} \) |
| 59 | \( 1 + (5.83e7 + 2.12e7i)T + (6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (-3.40e7 - 2.85e7i)T + (2.03e15 + 1.15e16i)T^{2} \) |
| 67 | \( 1 + (1.75e8 - 6.38e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (3.41e6 - 2.86e6i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-2.11e7 - 1.20e8i)T + (-5.53e16 + 2.01e16i)T^{2} \) |
| 79 | \( 1 + (-2.04e7 - 1.16e8i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (2.84e8 - 4.92e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (7.98e7 - 4.52e8i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (1.24e9 + 4.54e8i)T + (5.82e17 + 4.88e17i)T^{2} \) |
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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
−14.06576209476077331163625254534, −13.04045088236091058972420764401, −12.23625720806940433763138410749, −11.08454847771016610499934014273, −9.211034275335733038704199867755, −8.259579940014571180795831235084, −6.72668104110216192556092033203, −5.72216512943768579172867102129, −2.52023247119875528843021518825, −1.39435584286478749485907202770,
0.05849549895505681610206918619, 2.79012712668517476627352921971, 4.61177305069913757694180857893, 6.02330403311440593226496431630, 7.73114830539133304300932424501, 9.649045519179723134145791955729, 10.18257461458934251591232155705, 10.81584131302012023744209871353, 12.88969656446151671225920921290, 14.71966757047960258406821073547
