| L(s) = 1 | + (−0.909 + 1.57i)2-s + (−7.01 − 13.9i)3-s + (14.3 + 24.8i)4-s + (48.3 + 83.6i)5-s + (28.2 + 1.61i)6-s + (3.10 − 5.37i)7-s − 110.·8-s + (−144. + 195. i)9-s − 175.·10-s + (60.5 − 104. i)11-s + (245. − 374. i)12-s + (−97.0 − 168. i)13-s + (5.64 + 9.77i)14-s + (826. − 1.25e3i)15-s + (−358. + 621. i)16-s + 88.6·17-s + ⋯ |
| L(s) = 1 | + (−0.160 + 0.278i)2-s + (−0.449 − 0.893i)3-s + (0.448 + 0.776i)4-s + (0.864 + 1.49i)5-s + (0.320 + 0.0182i)6-s + (0.0239 − 0.0414i)7-s − 0.609·8-s + (−0.595 + 0.803i)9-s − 0.555·10-s + (0.150 − 0.261i)11-s + (0.491 − 0.749i)12-s + (−0.159 − 0.275i)13-s + (0.00769 + 0.0133i)14-s + (0.948 − 1.44i)15-s + (−0.350 + 0.606i)16-s + 0.0743·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.506022 + 1.17704i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.506022 + 1.17704i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.01 + 13.9i)T \) |
| 11 | \( 1 + (-60.5 + 104. i)T \) |
| good | 2 | \( 1 + (0.909 - 1.57i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-48.3 - 83.6i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-3.10 + 5.37i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 13 | \( 1 + (97.0 + 168. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 88.6T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.67e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-273. - 472. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.65e3 - 6.33e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.98e3 - 3.44e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (7.80e3 + 1.35e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (6.03e3 - 1.04e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-4.48e3 + 7.77e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 2.30e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (2.25e3 + 3.91e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.67e4 - 2.89e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.27e3 + 5.66e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 7.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.32e3 - 5.75e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.15e4 + 2.00e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 7.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (1.86e4 - 3.23e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.30173667790511810853691879666, −12.29930135149710078775399544622, −11.16164479560884400536336872815, −10.45000821266374067255409425299, −8.691707808618085505681148560066, −7.33671712118106192036846073833, −6.72692247253880385359850505789, −5.76781346540843309446147914304, −3.17078206509972462435679224755, −2.03365441807918687899781210579,
0.52113612194244503450086695694, 2.02247156157377073110936497728, 4.40733508153866050894079214280, 5.43393702171481169531190842724, 6.33398041317092472374304657198, 8.629442351426295311939778565655, 9.551524875531948592898126519478, 10.15233244535105812570689598743, 11.41629426788611927824632451718, 12.33351215570862227601121370543
