L(s) = 1 | + (−1.10 + 1.33i)3-s − 5-s − 4.67i·7-s + (−0.571 − 2.94i)9-s + 1.14i·11-s + 2.42i·13-s + (1.10 − 1.33i)15-s + 7.87i·17-s − 2.42·19-s + (6.24 + 5.14i)21-s − 4.62·23-s + 25-s + (4.56 + 2.48i)27-s + 6.48·29-s + 3.14i·31-s + ⋯ |
L(s) = 1 | + (−0.636 + 0.771i)3-s − 0.447·5-s − 1.76i·7-s + (−0.190 − 0.981i)9-s + 0.344i·11-s + 0.672i·13-s + (0.284 − 0.345i)15-s + 1.90i·17-s − 0.556·19-s + (1.36 + 1.12i)21-s − 0.965·23-s + 0.200·25-s + (0.878 + 0.477i)27-s + 1.20·29-s + 0.564i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.280710 + 0.544984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.280710 + 0.544984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.10 - 1.33i)T \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + 4.67iT - 7T^{2} \) |
| 11 | \( 1 - 1.14iT - 11T^{2} \) |
| 13 | \( 1 - 2.42iT - 13T^{2} \) |
| 17 | \( 1 - 7.87iT - 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 - 3.14iT - 31T^{2} \) |
| 37 | \( 1 - 6.83iT - 37T^{2} \) |
| 41 | \( 1 - 5.88iT - 41T^{2} \) |
| 43 | \( 1 + 6.61T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 + 9.05T + 53T^{2} \) |
| 59 | \( 1 - 6.48iT - 59T^{2} \) |
| 61 | \( 1 + 1.48iT - 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 2.52T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 + 6.77iT - 79T^{2} \) |
| 83 | \( 1 - 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 6.73iT - 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−10.39373601762932347615698087154, −9.859048197993910350574065847639, −8.612808541641886753526920490000, −7.83526869271780277605315643411, −6.71450397545037975584712442462, −6.23708983720908753308972550682, −4.67669579350199938016350021025, −4.22792449261609558978178324194, −3.46281265445240931795599005945, −1.35376000513481228759975009818,
0.32581222746375736969159998301, 2.16460412778715340718002778936, 2.99334970229130652985588089355, 4.72050408109971312484830045814, 5.53809974939559969517812749192, 6.16663616103188858835083725241, 7.19052699745288473903418492295, 8.080725308158701635638336652708, 8.727609171346884235874386757105, 9.670502911182542831648913828039