| L(s) = 1 | + (2 + 3.46i)2-s + (−7.29 + 13.7i)3-s + (−7.99 + 13.8i)4-s + (−6.92 + 11.9i)5-s + (−62.3 + 2.26i)6-s + (62.0 + 107. i)7-s − 63.9·8-s + (−136. − 201. i)9-s − 55.3·10-s + (−60.5 − 104. i)11-s + (−132. − 211. i)12-s + (−315. + 546. i)13-s + (−248. + 430. i)14-s + (−114. − 182. i)15-s + (−128 − 221. i)16-s − 1.19e3·17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.468 + 0.883i)3-s + (−0.249 + 0.433i)4-s + (−0.123 + 0.214i)5-s + (−0.706 + 0.0257i)6-s + (0.478 + 0.829i)7-s − 0.353·8-s + (−0.561 − 0.827i)9-s − 0.175·10-s + (−0.150 − 0.261i)11-s + (−0.265 − 0.423i)12-s + (−0.517 + 0.896i)13-s + (−0.338 + 0.586i)14-s + (−0.131 − 0.209i)15-s + (−0.125 − 0.216i)16-s − 1.00·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3215888982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3215888982\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 - 3.46i)T \) |
| 3 | \( 1 + (7.29 - 13.7i)T \) |
| 11 | \( 1 + (60.5 + 104. i)T \) |
| good | 5 | \( 1 + (6.92 - 11.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-62.0 - 107. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 13 | \( 1 + (315. - 546. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.19e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.52e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.19e3 + 2.06e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.12e3 + 1.95e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-949. + 1.64e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 839.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.67e3 + 4.63e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (4.42e3 + 7.66e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-4.36e3 - 7.55e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 2.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (5.00e3 - 8.66e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-337. - 583. i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-4.13e3 + 7.15e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.48e4 + 6.02e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.89e4 - 3.28e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 2.49e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.83e4 + 4.90e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.21126931747259213106936282058, −11.43406975521174878768006105677, −10.54428807771857753115178873422, −9.135392879797770903699052257762, −8.600096043355790818115702080583, −7.02148145975387016709188213196, −6.02014311797309282217306417065, −4.95707592954294840028367029002, −4.09845613949718657335547243811, −2.49351928986899798106928758760,
0.095326697728299391895908559230, 1.29258460007099489163780628956, 2.60636505602607978762779646070, 4.34377662076774138363391744392, 5.30467750780764377327623419507, 6.65190504062973482690266274690, 7.64832605120248503242556584604, 8.707382547336587625855803991429, 10.29606100220067821371726646657, 10.92830409297321613861054339697
