| L(s) = 1 | + (−0.342 + 0.939i)2-s + (−1.59 − 0.667i)3-s + (−0.766 − 0.642i)4-s + (0.105 − 0.599i)5-s + (1.17 − 1.27i)6-s + (−1.41 + 2.23i)7-s + (0.866 − 0.500i)8-s + (2.11 + 2.13i)9-s + (0.527 + 0.304i)10-s + (0.824 − 0.145i)11-s + (0.795 + 1.53i)12-s + (−1.82 − 5.02i)13-s + (−1.61 − 2.09i)14-s + (−0.569 + 0.888i)15-s + (0.173 + 0.984i)16-s + (1.99 − 3.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.922 − 0.385i)3-s + (−0.383 − 0.321i)4-s + (0.0472 − 0.268i)5-s + (0.479 − 0.520i)6-s + (−0.535 + 0.844i)7-s + (0.306 − 0.176i)8-s + (0.703 + 0.710i)9-s + (0.166 + 0.0962i)10-s + (0.248 − 0.0438i)11-s + (0.229 + 0.444i)12-s + (−0.507 − 1.39i)13-s + (−0.431 − 0.559i)14-s + (−0.146 + 0.229i)15-s + (0.0434 + 0.246i)16-s + (0.483 − 0.836i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.738794 - 0.177227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738794 - 0.177227i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 3 | \( 1 + (1.59 + 0.667i)T \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
| good | 5 | \( 1 + (-0.105 + 0.599i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.824 + 0.145i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.82 + 5.02i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.99 + 3.44i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.27 + 3.04i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.49 + 4.16i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.34 + 9.19i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (6.03 - 7.19i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (4.69 - 8.13i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.99 + 2.90i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.298 + 1.69i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.01 - 1.68i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 + (1.06 - 6.03i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.15 + 1.37i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.12 + 1.86i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.21 - 4.74i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.28 + 1.32i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.21 + 1.53i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (5.23 + 1.90i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.04 + 3.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.62 + 0.638i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.37280607948320536594591394941, −10.24467135857962508431509381958, −9.476317568834348850977394411473, −8.415839245982073126297613388988, −7.34497803414645409099484601620, −6.56578268102031059601411571513, −5.40303878382085079116609436362, −5.02461737795732742523668317746, −2.90788033199269302687051352359, −0.72389429884665458217321166856,
1.33117461695005676090141861187, 3.41016512951998973672842848155, 4.27024049990775477808509270066, 5.51581657365456340088775344423, 6.77441326151484224126704924115, 7.47291016063117792595183567741, 9.220782684869665500240202020725, 9.687528906907833872238926156007, 10.69491478778938432101311754665, 11.18676349501137991429499628580
