| L(s) = 1 | + (−0.939 − 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.0923 + 0.0775i)5-s + (−0.173 + 0.984i)6-s + (2.14 − 3.71i)7-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.113 − 0.0412i)10-s + (−1.28 − 2.22i)11-s + (0.5 − 0.866i)12-s + (0.141 − 0.802i)13-s + (−3.28 + 2.75i)14-s + (0.0923 + 0.0775i)15-s + (0.173 + 0.984i)16-s + (0.439 + 0.160i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.100 − 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.0413 + 0.0346i)5-s + (−0.0708 + 0.402i)6-s + (0.810 − 1.40i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (0.0358 − 0.0130i)10-s + (−0.388 − 0.672i)11-s + (0.144 − 0.249i)12-s + (0.0392 − 0.222i)13-s + (−0.878 + 0.737i)14-s + (0.0238 + 0.0200i)15-s + (0.0434 + 0.246i)16-s + (0.106 + 0.0388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.619525 - 0.480953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.619525 - 0.480953i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-3.16 - 2.99i)T \) |
| good | 5 | \( 1 + (0.0923 - 0.0775i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.14 + 3.71i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.28 + 2.22i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.141 + 0.802i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.439 - 0.160i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.25 - 3.57i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.20 - 0.802i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.67 - 4.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 + (-0.666 - 3.77i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.14 + 5.99i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-8.90 + 3.24i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-9.77 - 8.20i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (14.1 + 5.14i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.31 + 1.10i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (10.7 - 3.91i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-10.2 + 8.57i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.396 - 2.24i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.843 - 4.78i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.62 - 2.81i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.595 + 3.37i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.91 + 1.06i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.45230959854829573691615791496, −12.18493764230321947989164734603, −11.03835914140071286943252782522, −10.52047623345231245524686790996, −9.001840673430290952263931898576, −7.72747321777651010138643965543, −7.21277773121902388004798520111, −5.43732285535737125663664636832, −3.51291803244063787848342247146, −1.28242763175821312831686930381,
2.42425829885683215657809595872, 4.76631647826269197688479663170, 5.81581155854740865283464263021, 7.41832664270459930427835493549, 8.659600900942567377538324837667, 9.351957991826813976040531813542, 10.60420970862715516956986178869, 11.58621541390161116531908440299, 12.47244273274454624151187109763, 14.14818601883500928532608943880
