| L(s) = 1 | + (−2.88 − 1.66i)2-s + (−26.7 + 4.00i)3-s + (−26.4 − 45.8i)4-s + (166. + 96.0i)5-s + (83.7 + 32.9i)6-s + (157. + 304. i)7-s + 389. i·8-s + (696. − 214. i)9-s + (−319. − 554. i)10-s + (−750. + 433. i)11-s + (889. + 1.11e3i)12-s + 3.41e3·13-s + (54.6 − 1.14e3i)14-s + (−4.82e3 − 1.89e3i)15-s + (−1.04e3 + 1.80e3i)16-s + (−2.06e3 + 1.19e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.360 − 0.208i)2-s + (−0.988 + 0.148i)3-s + (−0.413 − 0.715i)4-s + (1.33 + 0.768i)5-s + (0.387 + 0.152i)6-s + (0.457 + 0.888i)7-s + 0.760i·8-s + (0.955 − 0.293i)9-s + (−0.319 − 0.554i)10-s + (−0.564 + 0.325i)11-s + (0.514 + 0.646i)12-s + 1.55·13-s + (0.0199 − 0.415i)14-s + (−1.43 − 0.562i)15-s + (−0.254 + 0.441i)16-s + (−0.420 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.981837 + 0.351816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.981837 + 0.351816i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (26.7 - 4.00i)T \) |
| 7 | \( 1 + (-157. - 304. i)T \) |
| good | 2 | \( 1 + (2.88 + 1.66i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (-166. - 96.0i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (750. - 433. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 3.41e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (2.06e3 - 1.19e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.29e3 - 2.24e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-6.49e3 - 3.74e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 2.45e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.02e4 - 1.76e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-3.39e4 + 5.88e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 8.76e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.12e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (1.05e5 + 6.11e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.45e5 - 8.37e4i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.39e5 + 8.02e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.99e4 + 3.45e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-3.69e4 - 6.39e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 9.29e3iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (7.80e4 + 1.35e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.16e5 - 2.02e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 8.72e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.82e5 - 1.05e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.18e6T + 8.32e11T^{2} \) |
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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
−17.57092072168741759407546412450, −15.69897661937748624910631020702, −14.43379104249514379008628075020, −13.06967932956724525539323604018, −11.10869547947453536117732298077, −10.37015921885757294563317998600, −9.029769290518067725232307891872, −6.25694393758566796844834322860, −5.32578453503094994540844992058, −1.71559517028066329068413388382,
0.955308097150233565553781510414, 4.64795125404776215419752749480, 6.34770378260261294891410016590, 8.214648399784020659666306923682, 9.784390181685032084731520772826, 11.22678560369360850283511754799, 13.24303247901064334414573700291, 13.34343406765518834582722307677, 16.09648874874367807690202612495, 16.95941553889843956845735512569
