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 | \( 1 + (2.83e4 - 4.90e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−10.92830409297321613861054339697, −10.29606100220067821371726646657, −8.707382547336587625855803991429, −7.64832605120248503242556584604, −6.65190504062973482690266274690, −5.30467750780764377327623419507, −4.34377662076774138363391744392, −2.60636505602607978762779646070, −1.29258460007099489163780628956, −0.095326697728299391895908559230,
2.49351928986899798106928758760, 4.09845613949718657335547243811, 4.95707592954294840028367029002, 6.02014311797309282217306417065, 7.02148145975387016709188213196, 8.600096043355790818115702080583, 9.135392879797770903699052257762, 10.54428807771857753115178873422, 11.43406975521174878768006105677, 12.21126931747259213106936282058