| L(s) = 1 | + (2.35 + 1.71i)2-s + (−2.48 + 7.65i)3-s + (0.155 + 0.477i)4-s + (−6.84 + 4.96i)5-s + (−18.9 + 13.7i)6-s + (2.16 + 6.65i)7-s + (6.75 − 20.7i)8-s + (−30.5 − 22.2i)9-s − 24.6·10-s + (−23.1 + 28.2i)11-s − 4.04·12-s + (41.3 + 30.0i)13-s + (−6.30 + 19.4i)14-s + (−21.0 − 64.7i)15-s + (54.8 − 39.8i)16-s + (−12.0 + 8.77i)17-s + ⋯ |
| L(s) = 1 | + (0.834 + 0.605i)2-s + (−0.478 + 1.47i)3-s + (0.0194 + 0.0597i)4-s + (−0.611 + 0.444i)5-s + (−1.29 + 0.938i)6-s + (0.116 + 0.359i)7-s + (0.298 − 0.918i)8-s + (−1.13 − 0.822i)9-s − 0.779·10-s + (−0.634 + 0.773i)11-s − 0.0972·12-s + (0.882 + 0.641i)13-s + (−0.120 + 0.370i)14-s + (−0.361 − 1.11i)15-s + (0.856 − 0.622i)16-s + (−0.172 + 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.271747 + 1.54086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.271747 + 1.54086i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.16 - 6.65i)T \) |
| 11 | \( 1 + (23.1 - 28.2i)T \) |
| good | 2 | \( 1 + (-2.35 - 1.71i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (2.48 - 7.65i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (6.84 - 4.96i)T + (38.6 - 118. i)T^{2} \) |
| 13 | \( 1 + (-41.3 - 30.0i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (12.0 - 8.77i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (19.0 - 58.7i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (11.3 + 35.0i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-213. - 155. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (46.8 + 144. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-130. + 402. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 30.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-164. + 505. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-71.0 - 51.5i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-73.1 - 225. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (650. - 472. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 218.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-588. + 427. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (1.82 + 5.61i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-317. - 230. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (495. - 359. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 558.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.00e3 - 732. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−14.94415206394278757331464218975, −13.70333740339700710695657806032, −12.28987041100127733561878196656, −10.99992129014268629071707931402, −10.23948890591357875690439253929, −8.962355739465794910562580738190, −7.11393346475127762565635571712, −5.71338084309841461165604269650, −4.67573471348486163908019180385, −3.67495185008468381137728699670,
0.866811132147569200367781861275, 2.92609705192286592030111225853, 4.72369504811951697633234888493, 6.15707841971688153004186586819, 7.69827244662739446604762998608, 8.435429628259223259475007328275, 10.97968668367550683493976588950, 11.49892899845087332285763597527, 12.68409216032642040739166946205, 13.15994383858849659040373791972
