| L(s) = 1 | + (1.61 + 1.17i)2-s + (2.64 − 8.12i)3-s + (1.23 + 3.80i)4-s + (−10.3 + 7.48i)5-s + (13.8 − 10.0i)6-s + (7.24 + 22.3i)7-s + (−2.47 + 7.60i)8-s + (−37.2 − 27.0i)9-s − 25.4·10-s + (−3.69 − 36.2i)11-s + 34.1·12-s + (−9.27 − 6.73i)13-s + (−14.4 + 44.6i)14-s + (33.6 + 103. i)15-s + (−12.9 + 9.40i)16-s + (52.9 − 38.5i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.508 − 1.56i)3-s + (0.154 + 0.475i)4-s + (−0.921 + 0.669i)5-s + (0.940 − 0.683i)6-s + (0.391 + 1.20i)7-s + (−0.109 + 0.336i)8-s + (−1.37 − 1.00i)9-s − 0.805·10-s + (−0.101 − 0.994i)11-s + 0.822·12-s + (−0.197 − 0.143i)13-s + (−0.276 + 0.851i)14-s + (0.578 + 1.78i)15-s + (−0.202 + 0.146i)16-s + (0.756 − 0.549i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.46851 - 0.139106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46851 - 0.139106i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 - 1.17i)T \) |
| 11 | \( 1 + (3.69 + 36.2i)T \) |
| good | 3 | \( 1 + (-2.64 + 8.12i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (10.3 - 7.48i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (-7.24 - 22.3i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (9.27 + 6.73i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-52.9 + 38.5i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (2.24 - 6.89i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (39.3 + 121. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-233. - 169. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (26.3 + 81.2i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (41.8 - 128. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (41.5 - 128. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (405. + 294. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-201. - 619. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (295. - 215. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 294.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (107. - 77.8i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-145. - 446. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (330. + 239. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-1.09e3 + 799. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 260.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.14e3 + 831. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.84720920597659943333311155277, −15.89907952099138465069666035170, −14.70335721688984096085401876945, −13.77275429335399216464026987839, −12.29778520273756866983536395200, −11.59280781308044609981474328437, −8.419504813096985549378070924759, −7.54063715403432348854756224300, −6.01124755013911372474612167340, −2.87928150513178740951327685236,
3.90984967770940474486673319475, 4.68985297667322811981123857566, 7.931538838213455941251650101620, 9.726698556761715444130582992579, 10.74991367147899407513172985048, 12.23816710967871922496994421460, 13.99733360190087296919988055990, 15.04729612948247917659406567298, 16.01658358390686157830831115439, 17.11678320107260991577745086854
