L(s) = 1 | + 4.67e10i·2-s + (−4.93e16 + 8.50e15i)3-s − 1.00e21·4-s − 4.28e24i·5-s + (−3.97e26 − 2.30e27i)6-s − 1.75e29·7-s + 8.16e30i·8-s + (2.35e33 − 8.38e32i)9-s + 2.00e35·10-s + 2.13e36i·11-s + (4.96e37 − 8.55e36i)12-s − 1.41e39·13-s − 8.21e39i·14-s + (3.64e40 + 2.11e41i)15-s − 1.56e42·16-s + 1.47e43i·17-s + ⋯ |
L(s) = 1 | + 1.36i·2-s + (−0.985 + 0.169i)3-s − 0.852·4-s − 1.47i·5-s + (−0.231 − 1.34i)6-s − 0.463·7-s + 0.201i·8-s + (0.942 − 0.334i)9-s + 2.00·10-s + 0.758i·11-s + (0.839 − 0.144i)12-s − 1.45·13-s − 0.630i·14-s + (0.250 + 1.45i)15-s − 1.12·16-s + 1.27i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{71}{2})\) |
\(\approx\) |
\(0.4418918309\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4418918309\) |
\(L(36)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.93e16 - 8.50e15i)T \) |
good | 2 | \( 1 - 4.67e10iT - 1.18e21T^{2} \) |
| 5 | \( 1 + 4.28e24iT - 8.47e48T^{2} \) |
| 7 | \( 1 + 1.75e29T + 1.43e59T^{2} \) |
| 11 | \( 1 - 2.13e36iT - 7.89e72T^{2} \) |
| 13 | \( 1 + 1.41e39T + 9.46e77T^{2} \) |
| 17 | \( 1 - 1.47e43iT - 1.35e86T^{2} \) |
| 19 | \( 1 + 5.25e44T + 3.25e89T^{2} \) |
| 23 | \( 1 - 2.67e47iT - 2.09e95T^{2} \) |
| 29 | \( 1 - 1.32e51iT - 2.33e102T^{2} \) |
| 31 | \( 1 - 2.33e52T + 2.48e104T^{2} \) |
| 37 | \( 1 + 9.09e54T + 5.94e109T^{2} \) |
| 41 | \( 1 - 1.39e56iT - 7.85e112T^{2} \) |
| 43 | \( 1 - 1.81e57T + 2.20e114T^{2} \) |
| 47 | \( 1 + 3.74e58iT - 1.11e117T^{2} \) |
| 53 | \( 1 + 1.66e60iT - 5.00e120T^{2} \) |
| 59 | \( 1 - 1.05e62iT - 9.11e123T^{2} \) |
| 61 | \( 1 + 2.28e62T + 9.39e124T^{2} \) |
| 67 | \( 1 + 1.70e63T + 6.68e127T^{2} \) |
| 71 | \( 1 + 3.89e64iT - 3.87e129T^{2} \) |
| 73 | \( 1 + 3.18e64T + 2.70e130T^{2} \) |
| 79 | \( 1 + 4.58e66T + 6.82e132T^{2} \) |
| 83 | \( 1 + 7.08e66iT - 2.16e134T^{2} \) |
| 89 | \( 1 + 2.22e68iT - 2.86e136T^{2} \) |
| 97 | \( 1 - 4.55e69T + 1.18e139T^{2} \) |
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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
−12.94561749365138742951810318228, −12.09758047202798109724372364258, −10.07674670981020257336038790121, −8.749004666437603396235830204769, −7.35777958933324116731515353044, −6.18899672537401654912630594023, −5.08357078084084578716055892105, −4.41847391659037067460456518388, −1.72220038318413873492385333913, −0.18437698104493462034866024652,
0.60588117942516907603900564035, 2.29702873400894913898060429606, 2.99929199588660383701646980630, 4.49874704709884210071752186863, 6.28501751709841966609725667624, 7.20677123593137991116810064598, 9.764362191632119401846225805385, 10.58444916098383907330574038345, 11.48696604096073735337048951453, 12.46869296252113724681115696842