| L(s) = 1 | + (3.90 − 2.51i)2-s + (−2.21 + 15.4i)3-s + (−17.6 + 38.5i)4-s + (7.91 + 26.9i)5-s + (30.0 + 65.8i)6-s + (210. − 182. i)7-s + (70.3 + 489. i)8-s + (−233. − 68.4i)9-s + (98.6 + 85.5i)10-s + (313. − 488. i)11-s + (−556. − 357. i)12-s + (−2.77e3 + 3.19e3i)13-s + (365. − 1.24e3i)14-s + (−433. + 62.3i)15-s + (−272. − 314. i)16-s + (−5.89e3 + 2.69e3i)17-s + ⋯ |
| L(s) = 1 | + (0.488 − 0.314i)2-s + (−0.0821 + 0.571i)3-s + (−0.275 + 0.602i)4-s + (0.0633 + 0.215i)5-s + (0.139 + 0.305i)6-s + (0.614 − 0.532i)7-s + (0.137 + 0.955i)8-s + (−0.319 − 0.0939i)9-s + (0.0986 + 0.0855i)10-s + (0.235 − 0.366i)11-s + (−0.321 − 0.206i)12-s + (−1.26 + 1.45i)13-s + (0.133 − 0.453i)14-s + (−0.128 + 0.0184i)15-s + (−0.0665 − 0.0768i)16-s + (−1.19 + 0.547i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.462560 + 1.26892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.462560 + 1.26892i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.21 - 15.4i)T \) |
| 23 | \( 1 + (9.58e3 - 7.49e3i)T \) |
| good | 2 | \( 1 + (-3.90 + 2.51i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-7.91 - 26.9i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (-210. + 182. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (-313. + 488. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (2.77e3 - 3.19e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (5.89e3 - 2.69e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (4.98e3 + 2.27e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (-1.02e4 - 2.24e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (3.43e3 + 2.38e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (1.91e4 - 6.53e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (-1.16e5 + 3.42e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (-6.88e4 - 9.89e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 + 1.83e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (9.65e4 - 8.36e4i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.67e5 + 1.92e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (-3.38e5 + 4.87e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (-2.49e5 - 3.88e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (-2.07e5 + 1.33e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-1.06e4 + 2.33e4i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-2.71e5 - 2.35e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-4.29e4 + 1.46e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (6.18e5 + 8.88e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (-3.99e5 - 1.36e6i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−14.10646093886824092803648449508, −12.83004248715868280002207450741, −11.63025695885127671917986006129, −10.89582008021224591540724960622, −9.385558331507227231056986759001, −8.240718841434932093162524199297, −6.74918711965674014222828740667, −4.77490657501202056039653553565, −4.05098799459308739185214448781, −2.29878401779489398846303047715,
0.43822576847891257935634327294, 2.23092078165169819899046080189, 4.60952771210011890016838120203, 5.58296564351938643621365936514, 6.88608357038923737553231327775, 8.274012766984657766495153084170, 9.624459698216192095652560171831, 10.88570031628776969573272223620, 12.38111222372270546961639347993, 13.00816839449280643249105600657
