| L(s) = 1 | + (−8.62 + 6.26i)2-s + (1.83 − 5.63i)3-s + (25.2 − 77.7i)4-s + (54.8 + 10.7i)5-s + (19.5 + 60.1i)6-s − 120.·7-s + (164. + 504. i)8-s + (168. + 122. i)9-s + (−540. + 251. i)10-s + (358. − 260. i)11-s + (−392. − 284. i)12-s + (721. + 524. i)13-s + (1.03e3 − 754. i)14-s + (161. − 289. i)15-s + (−2.46e3 − 1.79e3i)16-s + (148. + 456. i)17-s + ⋯ |
| L(s) = 1 | + (−1.52 + 1.10i)2-s + (0.117 − 0.361i)3-s + (0.789 − 2.43i)4-s + (0.981 + 0.192i)5-s + (0.221 + 0.681i)6-s − 0.928·7-s + (0.906 + 2.78i)8-s + (0.692 + 0.502i)9-s + (−1.71 + 0.794i)10-s + (0.892 − 0.648i)11-s + (−0.786 − 0.571i)12-s + (1.18 + 0.860i)13-s + (1.41 − 1.02i)14-s + (0.184 − 0.332i)15-s + (−2.40 − 1.74i)16-s + (0.124 + 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.769529 + 0.417921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.769529 + 0.417921i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-54.8 - 10.7i)T \) |
| good | 2 | \( 1 + (8.62 - 6.26i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-1.83 + 5.63i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 + 120.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-358. + 260. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-721. - 524. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-148. - 456. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-36.3 - 111. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.72e3 + 1.25e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-198. + 609. i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (501. + 1.54e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (9.78e3 + 7.10e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.01e4 - 7.39e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.17e3 - 1.90e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.14e4 + 3.51e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.29e4 - 9.42e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (3.49e4 - 2.53e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (5.58e3 + 1.71e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (8.08e3 - 2.48e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.22e4 - 8.87e3i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.35e4 + 4.16e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (7.29e3 + 2.24e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (3.95e4 - 2.87e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.32e3 - 1.32e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−16.66854615325369819082482807000, −16.10520581418227170341429339104, −14.48555738192777589990444173419, −13.35334633087235975282952256050, −10.84686514413652792142515873870, −9.646138896114961889941210560558, −8.663130141469031263386100090163, −6.86866928614502475510692624308, −6.11255735393849404622692254469, −1.42791520088258926772672887503,
1.26261958967453460085846795242, 3.36910861734671912703230599720, 6.81450729169299373220582363215, 8.899771376870180320335699813306, 9.665718621214295514190266968359, 10.54676139538534006810609228304, 12.24713978309673330845672945730, 13.25726196131445501135004350903, 15.62539923284820071198963666819, 16.80527697570418475301492347949
