L(s) = 1 | + (1.63 − 1.18i)2-s + (−0.809 − 2.49i)3-s + (0.647 − 1.99i)4-s + (−4.29 − 3.11i)6-s + (0.298 − 0.918i)7-s + (−0.0595 − 0.183i)8-s + (−3.12 + 2.27i)9-s + (−3.31 + 0.189i)11-s − 5.49·12-s + (3.66 − 2.65i)13-s + (−0.603 − 1.85i)14-s + (3.07 + 2.23i)16-s + (2.69 + 1.96i)17-s + (−2.41 + 7.44i)18-s + (1.01 + 3.11i)19-s + ⋯ |
L(s) = 1 | + (1.15 − 0.841i)2-s + (−0.467 − 1.43i)3-s + (0.323 − 0.996i)4-s + (−1.75 − 1.27i)6-s + (0.112 − 0.347i)7-s + (−0.0210 − 0.0648i)8-s + (−1.04 + 0.757i)9-s + (−0.998 + 0.0572i)11-s − 1.58·12-s + (1.01 − 0.737i)13-s + (−0.161 − 0.496i)14-s + (0.768 + 0.558i)16-s + (0.654 + 0.475i)17-s + (−0.570 + 1.75i)18-s + (0.232 + 0.715i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.632830 - 1.84204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.632830 - 1.84204i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + (3.31 - 0.189i)T \) |
good | 2 | \( 1 + (-1.63 + 1.18i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.809 + 2.49i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.298 + 0.918i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.66 + 2.65i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.69 - 1.96i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.01 - 3.11i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + (-1.51 + 4.67i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.338 + 0.245i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.95 + 6.02i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.78 - 5.50i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + (-1.33 - 4.11i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.15 + 1.56i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.12 - 9.62i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.99 - 1.45i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 9.60T + 67T^{2} \) |
| 71 | \( 1 + (-4.41 - 3.20i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.443 + 1.36i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.812 - 0.590i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.98 + 4.34i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + (-2.44 + 1.77i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.87408133961494299983365070371, −10.93961846685067281413221786610, −10.21475241176205291307325549511, −8.154016298530555672183671247874, −7.64951933533000909828904004373, −6.07586336647953913270150676694, −5.57560307468741288731835649773, −4.03033010007937574155635535886, −2.66871231753699609650735386782, −1.30367025921154396788616027944,
3.23316617100259924738788111020, 4.30332517529468013650245817438, 5.16801216453007885119829796792, 5.80518909109479622328173659078, 7.00037311118589519258632123271, 8.371999453372257984990945866770, 9.542094405218379472810446164616, 10.43527801724498256294824911229, 11.36233464780402505674761645503, 12.30875361669846668096535340780