L(s) = 1 | − 2-s + (1.71 − 0.272i)3-s + 4-s + (1.59 + 2.75i)5-s + (−1.71 + 0.272i)6-s + (−2.56 − 0.658i)7-s − 8-s + (2.85 − 0.931i)9-s + (−1.59 − 2.75i)10-s + (−1.59 + 2.75i)11-s + (1.71 − 0.272i)12-s + (2.85 − 4.93i)13-s + (2.56 + 0.658i)14-s + (3.47 + 4.28i)15-s + 16-s + (−0.760 − 1.31i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.987 − 0.157i)3-s + 0.5·4-s + (0.711 + 1.23i)5-s + (−0.698 + 0.111i)6-s + (−0.968 − 0.249i)7-s − 0.353·8-s + (0.950 − 0.310i)9-s + (−0.503 − 0.871i)10-s + (−0.479 + 0.830i)11-s + (0.493 − 0.0785i)12-s + (0.790 − 1.36i)13-s + (0.684 + 0.176i)14-s + (0.896 + 1.10i)15-s + 0.250·16-s + (−0.184 − 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06685 + 0.162213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06685 + 0.162213i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.71 + 0.272i)T \) |
| 7 | \( 1 + (2.56 + 0.658i)T \) |
good | 5 | \( 1 + (-1.59 - 2.75i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.59 - 2.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.760 + 1.31i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.641 - 1.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.11 + 1.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 9.42T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 + (-1.02 - 1.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 8.69T + 71T^{2} \) |
| 73 | \( 1 + (2.48 + 4.30i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 + (4.03 + 6.98i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.112 + 0.195i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.42 - 12.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.35903265525074886651046766527, −12.74398658146481157262066778976, −10.87706633430871348371638007932, −10.07768264473451158633234779635, −9.473391268741702771582022806324, −8.006248731667418759758946419130, −7.09042613394326803308586615805, −6.07947531546242995099900252362, −3.44125436550772866138974481360, −2.35854856132667627291828329605,
1.85617955315654329620368695849, 3.67301402463892483769100570211, 5.54941311319682606125326317940, 6.93677617308514214511416834780, 8.558379471628900280370049802263, 8.991248420725262629967408192999, 9.731785379738562141329801269055, 11.02414798299731951546670477714, 12.62201042515457502256080799295, 13.26923284259440935553396578644