L(s) = 1 | + (2.54 − 4.40i)2-s + (−1.04 − 1.80i)3-s + (3.08 + 5.33i)4-s + (−20.9 + 36.2i)5-s − 10.5·6-s + (−127. − 25.2i)7-s + 193.·8-s + (119. − 206. i)9-s + (106. + 184. i)10-s + (−36.0 − 62.4i)11-s + (6.42 − 11.1i)12-s − 632.·13-s + (−434. + 495. i)14-s + 87.1·15-s + (394. − 683. i)16-s + (987. + 1.71e3i)17-s + ⋯ |
L(s) = 1 | + (0.449 − 0.778i)2-s + (−0.0668 − 0.115i)3-s + (0.0963 + 0.166i)4-s + (−0.374 + 0.647i)5-s − 0.120·6-s + (−0.980 − 0.194i)7-s + 1.07·8-s + (0.491 − 0.850i)9-s + (0.336 + 0.582i)10-s + (−0.0898 − 0.155i)11-s + (0.0128 − 0.0222i)12-s − 1.03·13-s + (−0.592 + 0.675i)14-s + 0.0999·15-s + (0.385 − 0.667i)16-s + (0.829 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.16538 - 0.396739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16538 - 0.396739i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (127. + 25.2i)T \) |
good | 2 | \( 1 + (-2.54 + 4.40i)T + (-16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (1.04 + 1.80i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (20.9 - 36.2i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (36.0 + 62.4i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 632.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-987. - 1.71e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-932. + 1.61e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (206. - 358. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 731.T + 2.05e7T^{2} \) |
| 31 | \( 1 + (3.06e3 + 5.30e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (5.17e3 - 8.96e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 3.52e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.07e4 + 1.85e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (6.28e3 + 1.08e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.80e4 - 3.12e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.01e3 - 3.48e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (7.78e3 + 1.34e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 1.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (9.79e3 + 1.69e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.80e4 - 3.12e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-3.51e4 + 6.08e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.05e5T + 8.58e9T^{2} \) |
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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
−21.57308437615047078910039825640, −20.00320751269634992621199298293, −18.92289127552869610966216251665, −16.93141700154744777014958807001, −15.15826741594825591264719986097, −13.11981359757681676277581925418, −11.90327030397803109767832616100, −10.14284762309064775210492991809, −7.10655360964114683232862426342, −3.43551399195511712236903657816,
5.12553753782728504599967213955, 7.37732792484842868284117396695, 9.998802688611086109464520430262, 12.44501284768504795812830520917, 14.13425557388889850963185316499, 15.94156673157555611824397647322, 16.47192746263695183583764465299, 18.96717658600509364384564236085, 20.17980668027821707748323859189, 22.17624364857314182956509306672