L(s) = 1 | + (−1.18 + 2.04i)2-s + (4.45 − 2.57i)3-s + (−0.796 − 1.37i)4-s + (−1.93 − 1.11i)5-s + 12.1i·6-s + (−1.42 + 6.85i)7-s − 5.69·8-s + (8.72 − 15.1i)9-s + (4.57 − 2.64i)10-s + (−6.86 − 11.8i)11-s + (−7.09 − 4.09i)12-s + 6.14i·13-s + (−12.3 − 11.0i)14-s − 11.5·15-s + (9.91 − 17.1i)16-s + (−1.74 + 1.00i)17-s + ⋯ |
L(s) = 1 | + (−0.591 + 1.02i)2-s + (1.48 − 0.857i)3-s + (−0.199 − 0.344i)4-s + (−0.387 − 0.223i)5-s + 2.02i·6-s + (−0.203 + 0.979i)7-s − 0.711·8-s + (0.969 − 1.67i)9-s + (0.457 − 0.264i)10-s + (−0.624 − 1.08i)11-s + (−0.591 − 0.341i)12-s + 0.472i·13-s + (−0.882 − 0.787i)14-s − 0.766·15-s + (0.619 − 1.07i)16-s + (−0.102 + 0.0591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.997874 + 0.372992i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.997874 + 0.372992i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.93 + 1.11i)T \) |
| 7 | \( 1 + (1.42 - 6.85i)T \) |
good | 2 | \( 1 + (1.18 - 2.04i)T + (-2 - 3.46i)T^{2} \) |
| 3 | \( 1 + (-4.45 + 2.57i)T + (4.5 - 7.79i)T^{2} \) |
| 11 | \( 1 + (6.86 + 11.8i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 6.14iT - 169T^{2} \) |
| 17 | \( 1 + (1.74 - 1.00i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (2.08 + 1.20i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.02 - 13.9i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 15.9T + 841T^{2} \) |
| 31 | \( 1 + (-17.9 + 10.3i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (6.51 - 11.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 55.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 54.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-61.9 - 35.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (33.6 + 58.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-18.2 + 10.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.8 + 13.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (37.2 + 64.4i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 66.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (108. - 62.5i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.0 + 84.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 78.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-38.7 - 22.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 31.6iT - 9.40e3T^{2} \) |
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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
−16.15115225547608644693268225864, −15.41634750643584501017140297340, −14.33646751744319511750288682048, −13.12450318649797162887209471385, −11.90577481553849794158658955645, −9.274940883569209300947278853422, −8.466003432342307160343984198624, −7.68782949341252012548212798650, −6.20692470184162015125425576547, −2.93283753868484626913008327884,
2.65955873603944789062487385177, 4.13109165277238082418957881702, 7.53902559562379236838421407041, 8.874027093163730894896327483706, 10.15882544616813418280657533652, 10.54922505814661445300794235199, 12.48394428322147583735450823148, 13.92018798203375517813423252904, 15.05586112155304423935224705272, 15.86832825571677894617864867120