L(s) = 1 | + (3.19 − 0.856i)3-s + (0.672 + 2.13i)5-s + (−2.52 − 0.802i)7-s + (6.87 − 3.97i)9-s + (−1.05 + 1.82i)11-s + (−1.20 − 1.20i)13-s + (3.97 + 6.23i)15-s + (0.850 + 3.17i)17-s + (−2.36 − 4.09i)19-s + (−8.74 − 0.406i)21-s + (−4.00 − 1.07i)23-s + (−4.09 + 2.86i)25-s + (11.5 − 11.5i)27-s − 3.65i·29-s + (−1.63 − 0.946i)31-s + ⋯ |
L(s) = 1 | + (1.84 − 0.494i)3-s + (0.300 + 0.953i)5-s + (−0.952 − 0.303i)7-s + (2.29 − 1.32i)9-s + (−0.317 + 0.550i)11-s + (−0.334 − 0.334i)13-s + (1.02 + 1.61i)15-s + (0.206 + 0.770i)17-s + (−0.542 − 0.939i)19-s + (−1.90 − 0.0886i)21-s + (−0.835 − 0.223i)23-s + (−0.819 + 0.573i)25-s + (2.22 − 2.22i)27-s − 0.679i·29-s + (−0.294 − 0.169i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.10159 - 0.113461i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10159 - 0.113461i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.672 - 2.13i)T \) |
| 7 | \( 1 + (2.52 + 0.802i)T \) |
good | 3 | \( 1 + (-3.19 + 0.856i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (1.05 - 1.82i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.20 + 1.20i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.850 - 3.17i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.36 + 4.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.00 + 1.07i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 + (1.63 + 0.946i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.65 - 9.91i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 0.826iT - 41T^{2} \) |
| 43 | \( 1 + (-4.70 + 4.70i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.90 - 1.04i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.645 + 2.40i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.15 - 2.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.44 + 0.837i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.2 - 3.28i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + (12.6 - 3.37i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-8.29 + 4.78i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.47 + 3.47i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.86 - 4.96i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.02 - 1.02i)T - 97iT^{2} \) |
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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
−12.19742622294142462204587430786, −10.41817850989842148806611408022, −9.925040204062516101961456689553, −8.988448486137131029246025442680, −7.88381356046880274943153937556, −7.13200961196635989229467311034, −6.30141365127582634878381240552, −4.07849723111300497225533374625, −3.03629247312622534167592108589, −2.17244108221938078542573524652,
2.08636153263751173857519947126, 3.29988769797313915342866553707, 4.34138868481999455482394011280, 5.71914317396105203535537090148, 7.34301914289298877771199249782, 8.309325808198395658343879622400, 9.101562808931187177769646223550, 9.604155918603207890424430811496, 10.49366372898800945671152682183, 12.30100085423007211226819591144