| L(s) = 1 | + (0.366 − 0.266i)2-s + (−1.58 − 4.88i)3-s + (−2.40 + 7.41i)4-s + (13.9 + 10.1i)5-s + (−1.88 − 1.36i)6-s + (2.16 − 6.65i)7-s + (2.21 + 6.80i)8-s + (0.494 − 0.359i)9-s + 7.81·10-s + (15.8 + 32.8i)11-s + 40.0·12-s + (60.0 − 43.5i)13-s + (−0.979 − 3.01i)14-s + (27.3 − 84.2i)15-s + (−47.8 − 34.7i)16-s + (10.2 + 7.47i)17-s + ⋯ |
| L(s) = 1 | + (0.129 − 0.0941i)2-s + (−0.305 − 0.940i)3-s + (−0.301 + 0.926i)4-s + (1.24 + 0.907i)5-s + (−0.128 − 0.0930i)6-s + (0.116 − 0.359i)7-s + (0.0977 + 0.300i)8-s + (0.0183 − 0.0133i)9-s + 0.247·10-s + (0.433 + 0.901i)11-s + 0.963·12-s + (1.28 − 0.930i)13-s + (−0.0187 − 0.0575i)14-s + (0.471 − 1.45i)15-s + (−0.747 − 0.542i)16-s + (0.146 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.71624 + 0.0949175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71624 + 0.0949175i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-15.8 - 32.8i)T \) |
| good | 2 | \( 1 + (-0.366 + 0.266i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (1.58 + 4.88i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-13.9 - 10.1i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (-60.0 + 43.5i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-10.2 - 7.47i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-28.6 - 88.0i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 83.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-9.45 + 29.1i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (188. - 136. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-117. + 360. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-59.0 - 181. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 215.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (58.1 + 178. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-384. + 279. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-83.7 + 257. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (725. + 527. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (232. + 168. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (166. - 513. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-539. + 392. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-480. - 348. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 61.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + (45.4 - 33.0i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.71350631171452039242629457391, −12.98749226009039097260366769967, −12.11334470837862195316996580691, −10.74427133258196462138426558998, −9.602695023124352042296401751127, −7.954061360222092098307479633504, −6.91922636046164415506865703749, −5.83718318912796231513307872043, −3.63683768381556505569397850600, −1.80228452581801810168554670425,
1.42493574187207581168678789862, 4.32568000619126751983980863738, 5.44338498947821277109292210149, 6.21264393414568997277182770353, 8.921837326767527714270661204204, 9.352633513171238309799913857303, 10.47891532090264380374978967125, 11.50966565589450509425146019940, 13.41137842577893803871920977001, 13.72940521801911211201777572484
