| L(s) = 1 | + (−3.87 + 4.12i)2-s + 18.2·3-s + (−2.00 − 31.9i)4-s + (−70.8 + 75.3i)6-s + 2.25i·7-s + (139. + 115. i)8-s + 91.3·9-s + 419. i·11-s + (−36.6 − 583. i)12-s + 106.·13-s + (−9.30 − 8.73i)14-s + (−1.01e3 + 127. i)16-s − 849. i·17-s + (−353. + 376. i)18-s + 335. i·19-s + ⋯ |
| L(s) = 1 | + (−0.684 + 0.728i)2-s + 1.17·3-s + (−0.0625 − 0.998i)4-s + (−0.803 + 0.854i)6-s + 0.0173i·7-s + (0.770 + 0.637i)8-s + 0.375·9-s + 1.04i·11-s + (−0.0734 − 1.17i)12-s + 0.175·13-s + (−0.0126 − 0.0119i)14-s + (−0.992 + 0.124i)16-s − 0.712i·17-s + (−0.257 + 0.273i)18-s + 0.213i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.748936853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748936853\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.87 - 4.12i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 18.2T + 243T^{2} \) |
| 7 | \( 1 - 2.25iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 419. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 106.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 849. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 335. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.54e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.20e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 61.9T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.27e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.34e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.34e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 6.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.12e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e5iT - 8.58e9T^{2} \) |
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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
−11.80623882411953050216714869229, −10.52444536941850391701623867454, −9.470229599496823795802439142566, −8.979899407503517679321127699996, −7.78748229865208752314679041548, −7.23268057688480453584252795483, −5.79161525250931331495957751071, −4.42827878190827558835473356704, −2.76167325539945230337783813830, −1.41110812247419322215631489183,
0.59521633487709350141530689228, 2.20059524333434645640328131994, 3.15026141554560566253275711032, 4.22477650312082011786470813835, 6.25276012582467654539530878303, 7.76293779531959761338710414003, 8.455313306648328450451045385518, 9.104031937598414937963639832760, 10.22150620321435482775091823815, 11.10037973258859169704634939916
