| L(s) = 1 | + (−2.77 + 4.93i)2-s + (−1.48 + 2.56i)3-s + (−16.6 − 27.3i)4-s + (−49.5 + 85.9i)5-s + (−8.55 − 14.4i)6-s + 147. i·7-s + (180. − 6.41i)8-s + (117. + 202. i)9-s + (−286. − 482. i)10-s − 472. i·11-s + (94.7 − 2.23i)12-s + (−652. + 376. i)13-s + (−728. − 408. i)14-s + (−146. − 254. i)15-s + (−469. + 909. i)16-s + (322. − 558. i)17-s + ⋯ |
| L(s) = 1 | + (−0.489 + 0.871i)2-s + (−0.0950 + 0.164i)3-s + (−0.520 − 0.853i)4-s + (−0.887 + 1.53i)5-s + (−0.0969 − 0.163i)6-s + 1.13i·7-s + (0.999 − 0.0354i)8-s + (0.481 + 0.834i)9-s + (−0.905 − 1.52i)10-s − 1.17i·11-s + (0.190 − 0.00448i)12-s + (−1.07 + 0.617i)13-s + (−0.992 − 0.557i)14-s + (−0.168 − 0.292i)15-s + (−0.458 + 0.888i)16-s + (0.270 − 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.267480 - 0.329399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267480 - 0.329399i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.77 - 4.93i)T \) |
| 19 | \( 1 + (305. + 1.54e3i)T \) |
| good | 3 | \( 1 + (1.48 - 2.56i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (49.5 - 85.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 147. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 472. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (652. - 376. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-322. + 558. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (256. - 148. i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (155. - 90.0i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 - 8.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.78e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (7.67e3 + 4.43e3i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-8.29e3 - 4.79e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.12e4 - 6.51e3i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.53e4 - 8.84e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (444. - 770. i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.48e4 + 2.56e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (6.11e3 + 1.05e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.41e4 - 2.45e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (3.05e4 - 5.28e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.67e3 - 2.90e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 6.26e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (-4.38e4 + 2.53e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-1.20e5 - 6.95e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−14.59013221703036177340863223412, −13.69111425400918247941089115463, −11.77647860303562107201245670837, −10.91038424858475753902258671840, −9.779786645003162064478614974820, −8.367625463113753737456487604108, −7.33628668223133011871480644601, −6.29677205844784474931695044515, −4.76441988446314114534826995780, −2.70883068241594072763620849956,
0.23913204460439034634883680289, 1.35809492050734124288947367056, 3.88106458497569621645152953830, 4.68615394477874983018567040548, 7.34089551221069664905938725115, 8.121110745274502077481295319759, 9.531220410520342541858523357451, 10.32505176304924279884437524780, 12.09526259235239983853628467941, 12.33819788145481874602572423309
