| L(s) = 1 | + (−6.12 − 5.14i)2-s + (−69.2 + 25.2i)3-s + (11.1 + 63.0i)4-s + (92.1 − 522. i)5-s + (553. + 201. i)6-s + (−577. − 1.00e3i)7-s + (256. − 443. i)8-s + (2.48e3 − 2.08e3i)9-s + (−3.25e3 + 2.73e3i)10-s + (−2.29e3 + 3.97e3i)11-s + (−2.35e3 − 4.08e3i)12-s + (−3.42e3 − 1.24e3i)13-s + (−1.60e3 + 9.09e3i)14-s + (6.79e3 + 3.85e4i)15-s + (−3.84e3 + 1.40e3i)16-s + (−1.26e4 − 1.06e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−1.48 + 0.538i)3-s + (0.0868 + 0.492i)4-s + (0.329 − 1.87i)5-s + (1.04 + 0.381i)6-s + (−0.636 − 1.10i)7-s + (0.176 − 0.306i)8-s + (1.13 − 0.953i)9-s + (−1.02 + 0.863i)10-s + (−0.519 + 0.900i)11-s + (−0.393 − 0.682i)12-s + (−0.432 − 0.157i)13-s + (−0.156 + 0.886i)14-s + (0.519 + 2.94i)15-s + (−0.234 + 0.0855i)16-s + (−0.625 − 0.525i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0711 - 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.000952474 + 0.00102280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000952474 + 0.00102280i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.12 + 5.14i)T \) |
| 19 | \( 1 + (-2.91e4 + 6.83e3i)T \) |
| good | 3 | \( 1 + (69.2 - 25.2i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-92.1 + 522. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (577. + 1.00e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.29e3 - 3.97e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.42e3 + 1.24e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.26e4 + 1.06e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-8.08e3 - 4.58e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (1.80e5 - 1.51e5i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-6.53e4 - 1.13e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 4.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-4.29e5 + 1.56e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-1.38e4 + 7.86e4i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (3.13e5 - 2.63e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (1.11e5 + 6.31e5i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-3.07e5 - 2.58e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-3.46e4 - 1.96e5i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (8.35e5 - 7.01e5i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (3.43e5 - 1.95e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (8.59e5 - 3.12e5i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (5.86e6 - 2.13e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (1.83e6 + 3.18e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-2.96e5 - 1.07e5i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-5.29e6 - 4.44e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−15.94362322979922182361483675789, −13.24511394696071000399157459690, −12.54880432683408706188464139992, −11.38475273624759064595963077681, −10.03737123276769583740689952346, −9.357802037679928987816325777565, −7.32423921713692546324806665004, −5.36424822674042566683264203730, −4.38375652756179418953242531058, −1.07582082099158150681002047769,
0.00105342875108454227703019808, 2.52674521648091759781888551027, 5.89287139539484151089794522863, 6.22769604701152744417262073948, 7.53604431555208436696386430450, 9.710363148231069858099780278646, 10.90470466127929407753363897418, 11.62134235919037655395372461637, 13.22144483959742194010893449013, 14.75416461484015846581174134167
