| L(s) = 1 | + (0.902 − 3.95i)2-s + (5.51 − 2.65i)3-s + (−7.60 − 3.66i)4-s + (−3.78 + 16.5i)5-s + (−5.51 − 24.1i)6-s + (−10.1 + 4.90i)7-s + (−1.11 + 1.39i)8-s + (6.50 − 8.15i)9-s + (62.0 + 29.9i)10-s + (−24.6 − 30.9i)11-s − 51.6·12-s + (31.3 + 39.2i)13-s + (10.1 + 44.6i)14-s + (23.1 + 101. i)15-s + (−37.5 − 47.1i)16-s + 96.3·17-s + ⋯ |
| L(s) = 1 | + (0.318 − 1.39i)2-s + (1.06 − 0.510i)3-s + (−0.950 − 0.457i)4-s + (−0.338 + 1.48i)5-s + (−0.375 − 1.64i)6-s + (−0.549 + 0.264i)7-s + (−0.0491 + 0.0616i)8-s + (0.240 − 0.301i)9-s + (1.96 + 0.945i)10-s + (−0.675 − 0.847i)11-s − 1.24·12-s + (0.668 + 0.837i)13-s + (0.194 + 0.852i)14-s + (0.398 + 1.74i)15-s + (−0.587 − 0.736i)16-s + 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0114 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0114 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.17304 - 1.15964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17304 - 1.15964i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-152. - 33.5i)T \) |
| good | 2 | \( 1 + (-0.902 + 3.95i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-5.51 + 2.65i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (3.78 - 16.5i)T + (-112. - 54.2i)T^{2} \) |
| 7 | \( 1 + (10.1 - 4.90i)T + (213. - 268. i)T^{2} \) |
| 11 | \( 1 + (24.6 + 30.9i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-31.3 - 39.2i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 - 96.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + (99.0 + 47.7i)T + (4.27e3 + 5.36e3i)T^{2} \) |
| 23 | \( 1 + (28.8 + 126. i)T + (-1.09e4 + 5.27e3i)T^{2} \) |
| 31 | \( 1 + (27.7 - 121. i)T + (-2.68e4 - 1.29e4i)T^{2} \) |
| 37 | \( 1 + (111. - 140. i)T + (-1.12e4 - 4.93e4i)T^{2} \) |
| 41 | \( 1 - 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (48.2 + 211. i)T + (-7.16e4 + 3.44e4i)T^{2} \) |
| 47 | \( 1 + (-105. - 132. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-108. + 475. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 - 469.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-292. + 140. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 + (-283. + 354. i)T + (-6.69e4 - 2.93e5i)T^{2} \) |
| 71 | \( 1 + (141. + 177. i)T + (-7.96e4 + 3.48e5i)T^{2} \) |
| 73 | \( 1 + (-118. - 517. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + (141. - 177. i)T + (-1.09e5 - 4.80e5i)T^{2} \) |
| 83 | \( 1 + (604. + 291. i)T + (3.56e5 + 4.47e5i)T^{2} \) |
| 89 | \( 1 + (-78.0 + 341. i)T + (-6.35e5 - 3.05e5i)T^{2} \) |
| 97 | \( 1 + (1.30e3 + 627. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−16.08070122740492202907561876592, −14.50119249463377176109474338677, −13.78190575558499587633634845659, −12.59423247265050411838602066695, −11.18472109519223707101281413452, −10.26247867035841338322808284369, −8.480620985048493133304772078599, −6.78070406424462703275285197135, −3.44230410685002501095312884020, −2.54546383022382430511368451092,
4.05525324962711702691160456552, 5.59472603113029564261432428827, 7.78092759647238393414303747012, 8.524251288873032838892867996580, 9.927143580413160653195141337649, 12.57037807282240298125442809354, 13.52975385607848631510281600663, 14.85386605782739734231249858704, 15.69271401965109758072340467137, 16.42312029829045509101777229254
