| L(s) = 1 | + (2.84 − 0.271i)2-s + (2.52 − 1.62i)3-s + (0.137 − 0.0265i)4-s + (1.10 − 7.66i)5-s + (6.72 − 5.29i)6-s + (−17.6 − 24.8i)7-s + (−21.5 + 6.31i)8-s + (3.73 − 8.18i)9-s + (1.05 − 22.0i)10-s + (11.3 + 8.93i)11-s + (0.304 − 0.290i)12-s + (1.07 + 4.41i)13-s + (−57.0 − 65.7i)14-s + (−9.65 − 21.1i)15-s + (−60.4 + 24.1i)16-s + (−124. − 23.9i)17-s + ⋯ |
| L(s) = 1 | + (1.00 − 0.0958i)2-s + (0.485 − 0.312i)3-s + (0.0172 − 0.00332i)4-s + (0.0985 − 0.685i)5-s + (0.457 − 0.360i)6-s + (−0.955 − 1.34i)7-s + (−0.950 + 0.279i)8-s + (0.138 − 0.303i)9-s + (0.0332 − 0.698i)10-s + (0.311 + 0.244i)11-s + (0.00733 − 0.00699i)12-s + (0.0228 + 0.0941i)13-s + (−1.08 − 1.25i)14-s + (−0.166 − 0.363i)15-s + (−0.944 + 0.378i)16-s + (−1.77 − 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.994892 - 1.96125i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.994892 - 1.96125i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.52 + 1.62i)T \) |
| 67 | \( 1 + (427. + 343. i)T \) |
| good | 2 | \( 1 + (-2.84 + 0.271i)T + (7.85 - 1.51i)T^{2} \) |
| 5 | \( 1 + (-1.10 + 7.66i)T + (-119. - 35.2i)T^{2} \) |
| 7 | \( 1 + (17.6 + 24.8i)T + (-112. + 324. i)T^{2} \) |
| 11 | \( 1 + (-11.3 - 8.93i)T + (313. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-1.07 - 4.41i)T + (-1.95e3 + 1.00e3i)T^{2} \) |
| 17 | \( 1 + (124. + 23.9i)T + (4.56e3 + 1.82e3i)T^{2} \) |
| 19 | \( 1 + (-63.5 + 89.1i)T + (-2.24e3 - 6.48e3i)T^{2} \) |
| 23 | \( 1 + (-118. - 61.0i)T + (7.05e3 + 9.91e3i)T^{2} \) |
| 29 | \( 1 + (-143. + 248. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (24.6 - 101. i)T + (-2.64e4 - 1.36e4i)T^{2} \) |
| 37 | \( 1 + (1.20 + 2.08i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (72.9 + 210. i)T + (-5.41e4 + 4.26e4i)T^{2} \) |
| 43 | \( 1 + (-240. + 278. i)T + (-1.13e4 - 7.86e4i)T^{2} \) |
| 47 | \( 1 + (-14.9 - 314. i)T + (-1.03e5 + 9.86e3i)T^{2} \) |
| 53 | \( 1 + (-193. - 223. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (399. - 117. i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-152. + 120. i)T + (5.35e4 - 2.20e5i)T^{2} \) |
| 71 | \( 1 + (75.2 - 14.5i)T + (3.32e5 - 1.33e5i)T^{2} \) |
| 73 | \( 1 + (11.7 - 9.25i)T + (9.17e4 - 3.78e5i)T^{2} \) |
| 79 | \( 1 + (196. - 186. i)T + (2.34e4 - 4.92e5i)T^{2} \) |
| 83 | \( 1 + (-180. + 72.3i)T + (4.13e5 - 3.94e5i)T^{2} \) |
| 89 | \( 1 + (-1.40e3 - 903. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-289. - 501. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.04649922985946935716902582907, −10.85470899911035474379382068364, −9.381266329098906822048237481117, −8.928778886873344462235618558961, −7.27182860642282123053197263963, −6.47366623655705159044804360860, −4.86131667594540265898463774831, −4.04841723372343134184594252330, −2.82051254986321096329852546477, −0.64312214467800277346623656350,
2.65182478499983062685111564009, 3.41324176212024075950739881446, 4.81928022751299324858695737500, 6.03528559457166952129503229965, 6.75495848335771106812534418491, 8.690802464153160530529625671517, 9.185086280641927479172853706110, 10.36419309151565993981604018721, 11.61234226619278418579959315535, 12.66001046697964230161616494577
