L(s) = 1 | + (−3.56 + 0.401i)2-s + (1.78 − 1.12i)3-s + (8.62 − 1.96i)4-s + (5.24 + 4.17i)5-s + (−5.90 + 4.70i)6-s + (2.11 − 9.28i)7-s + (−16.3 + 5.73i)8-s + (−1.97 + 4.11i)9-s + (−20.3 − 12.7i)10-s + (−0.700 − 0.245i)11-s + (13.1 − 13.1i)12-s + (3.39 + 7.04i)13-s + (−3.82 + 33.9i)14-s + (14.0 + 1.58i)15-s + (24.1 − 11.6i)16-s + (−17.4 − 17.4i)17-s + ⋯ |
L(s) = 1 | + (−1.78 + 0.200i)2-s + (0.594 − 0.373i)3-s + (2.15 − 0.492i)4-s + (1.04 + 0.835i)5-s + (−0.983 + 0.784i)6-s + (0.302 − 1.32i)7-s + (−2.04 + 0.716i)8-s + (−0.219 + 0.456i)9-s + (−2.03 − 1.27i)10-s + (−0.0637 − 0.0222i)11-s + (1.09 − 1.09i)12-s + (0.261 + 0.542i)13-s + (−0.273 + 2.42i)14-s + (0.935 + 0.105i)15-s + (1.51 − 0.727i)16-s + (−1.02 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.623147 + 0.00544588i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.623147 + 0.00544588i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (19.7 - 21.2i)T \) |
good | 2 | \( 1 + (3.56 - 0.401i)T + (3.89 - 0.890i)T^{2} \) |
| 3 | \( 1 + (-1.78 + 1.12i)T + (3.90 - 8.10i)T^{2} \) |
| 5 | \( 1 + (-5.24 - 4.17i)T + (5.56 + 24.3i)T^{2} \) |
| 7 | \( 1 + (-2.11 + 9.28i)T + (-44.1 - 21.2i)T^{2} \) |
| 11 | \( 1 + (0.700 + 0.245i)T + (94.6 + 75.4i)T^{2} \) |
| 13 | \( 1 + (-3.39 - 7.04i)T + (-105. + 132. i)T^{2} \) |
| 17 | \( 1 + (17.4 + 17.4i)T + 289iT^{2} \) |
| 19 | \( 1 + (7.96 - 12.6i)T + (-156. - 325. i)T^{2} \) |
| 23 | \( 1 + (11.9 + 15.0i)T + (-117. + 515. i)T^{2} \) |
| 31 | \( 1 + (-0.327 + 0.0369i)T + (936. - 213. i)T^{2} \) |
| 37 | \( 1 + (15.2 - 5.34i)T + (1.07e3 - 853. i)T^{2} \) |
| 41 | \( 1 + (-28.0 + 28.0i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-0.679 + 6.02i)T + (-1.80e3 - 411. i)T^{2} \) |
| 47 | \( 1 + (-1.43 + 4.10i)T + (-1.72e3 - 1.37e3i)T^{2} \) |
| 53 | \( 1 + (37.6 - 47.1i)T + (-625. - 2.73e3i)T^{2} \) |
| 59 | \( 1 - 91.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-6.43 + 4.04i)T + (1.61e3 - 3.35e3i)T^{2} \) |
| 67 | \( 1 + (-29.8 + 62.0i)T + (-2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (38.1 + 79.2i)T + (-3.14e3 + 3.94e3i)T^{2} \) |
| 73 | \( 1 + (29.1 + 3.28i)T + (5.19e3 + 1.18e3i)T^{2} \) |
| 79 | \( 1 + (-7.28 - 20.8i)T + (-4.87e3 + 3.89e3i)T^{2} \) |
| 83 | \( 1 + (-9.19 - 40.2i)T + (-6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (-98.0 + 11.0i)T + (7.72e3 - 1.76e3i)T^{2} \) |
| 97 | \( 1 + (-136. - 86.0i)T + (4.08e3 + 8.47e3i)T^{2} \) |
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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
−17.20381420185940539141020432792, −16.25641486207767262262849749484, −14.43066149536923724815595972320, −13.63106037057178340601902540659, −11.02591209877167334834463414520, −10.33181459063392946598500960859, −9.004828155093114908088000586703, −7.60532482196887026663776969558, −6.63320842727198425524264247209, −2.09147568901355725236710768500,
2.17403251881833238533724983390, 5.99760421440243241711211321872, 8.403278286763391354245687720317, 8.972834492986334023686666700909, 9.876193411912551826500286347404, 11.46016721054536833326264770882, 12.96155709527904930107817240449, 15.01496316256726878822835922114, 15.90876703702229667732650243733, 17.53534610553738230695584125403