L(s) = 1 | + (−1 + i)2-s − 2i·4-s + 3.46·5-s + (1.73 − 2i)7-s + (2 + 2i)8-s + (−3.46 + 3.46i)10-s + 2·11-s − 3.46·13-s + (0.267 + 3.73i)14-s − 4·16-s − 3.46i·17-s − 6.92i·19-s − 6.92i·20-s + (−2 + 2i)22-s − 2i·23-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − i·4-s + 1.54·5-s + (0.654 − 0.755i)7-s + (0.707 + 0.707i)8-s + (−1.09 + 1.09i)10-s + 0.603·11-s − 0.960·13-s + (0.0716 + 0.997i)14-s − 16-s − 0.840i·17-s − 1.58i·19-s − 1.54i·20-s + (−0.426 + 0.426i)22-s − 0.417i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36881 + 0.0490750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36881 + 0.0490750i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 6.92iT - 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 - 6.92iT - 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−10.72497783049281850545557290789, −9.683361077515018822229111935078, −9.386335865045905566752909678869, −8.300742647150131084746364330776, −7.03997022091557205765292905236, −6.67591707609765622212713724613, −5.30049183317735059237189828760, −4.75337781209501521388492084182, −2.46995466914332457304949869819, −1.18378488382907941152311842077,
1.70383546586965223269484873836, 2.30565822536694976416173352303, 3.92564134664087775194198994523, 5.38380406918337699017237087153, 6.20389807758513087279361355351, 7.53258055365310709124955844104, 8.494285440155704110633308375343, 9.359496625172534146355845992038, 9.920550227618452044710102595194, 10.69468602219105931231193044533