| L(s) = 1 | − 4.69i·3-s + (−23.4 + 50.7i)5-s − 10.2i·7-s + 220.·9-s − 596.·11-s + 420. i·13-s + (238. + 109. i)15-s − 974. i·17-s − 380.·19-s − 48.1·21-s + 3.54e3i·23-s + (−2.02e3 − 2.37e3i)25-s − 2.17e3i·27-s + 5.44e3·29-s − 3.62e3·31-s + ⋯ |
| L(s) = 1 | − 0.301i·3-s + (−0.419 + 0.907i)5-s − 0.0791i·7-s + 0.909·9-s − 1.48·11-s + 0.690i·13-s + (0.273 + 0.126i)15-s − 0.817i·17-s − 0.241·19-s − 0.0238·21-s + 1.39i·23-s + (−0.648 − 0.760i)25-s − 0.574i·27-s + 1.20·29-s − 0.677·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6733017396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6733017396\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (23.4 - 50.7i)T \) |
| good | 3 | \( 1 + 4.69iT - 243T^{2} \) |
| 7 | \( 1 + 10.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 596.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 420. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 974. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 380.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.54e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.75e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 263.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.44e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.34e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.90e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.16e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 8.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.51e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.29e4iT - 8.58e9T^{2} \) |
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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
−10.44890719774417605822813818525, −9.832337153291435846007941104182, −8.434419199142068710860258973928, −7.32090204813189579533585293135, −7.00958516277033845899059903322, −5.56460539700628240225327044176, −4.33623411511461873003633969814, −3.09287190966597809055576644506, −1.92706646031058121637840411138, −0.18948582447348967961414681821,
1.14687646472500973045576136169, 2.73574439866033537836486306785, 4.22053049540058034671384162238, 4.92764387936026587973165125001, 6.06459868227362595575809709827, 7.55695893516215936519822766378, 8.194790992081249211327176441642, 9.180391542920595754994519583565, 10.36847371849671746156380159455, 10.76653869529859718537887301479
