| L(s) = 1 | + (−1.5 + 2.59i)3-s + (10.6 + 18.5i)5-s + (−18.2 + 3.19i)7-s + (−4.5 − 7.79i)9-s + (−5.35 + 9.26i)11-s + 8.31·13-s − 64.1·15-s + (−0.742 + 1.28i)17-s + (−80.7 − 139. i)19-s + (19.0 − 52.1i)21-s + (61.9 + 107. i)23-s + (−166. + 288. i)25-s + 27·27-s − 222.·29-s + (−101. + 176. i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.956 + 1.65i)5-s + (−0.984 + 0.172i)7-s + (−0.166 − 0.288i)9-s + (−0.146 + 0.254i)11-s + 0.177·13-s − 1.10·15-s + (−0.0105 + 0.0183i)17-s + (−0.975 − 1.68i)19-s + (0.198 − 0.542i)21-s + (0.561 + 0.973i)23-s + (−1.33 + 2.30i)25-s + 0.192·27-s − 1.42·29-s + (−0.590 + 1.02i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.233588 + 1.11381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.233588 + 1.11381i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 2.59i)T \) |
| 7 | \( 1 + (18.2 - 3.19i)T \) |
| good | 5 | \( 1 + (-10.6 - 18.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (5.35 - 9.26i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 8.31T + 2.19e3T^{2} \) |
| 17 | \( 1 + (0.742 - 1.28i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (80.7 + 139. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-61.9 - 107. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (101. - 176. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-170. - 295. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 51.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (134. + 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-216. + 374. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (15.3 - 26.6i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-296. - 513. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (207. - 358. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 294.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-297. + 514. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-415. - 720. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 326.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-630. - 1.09e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−13.01780214151644248080023148433, −11.39419044149627106470703947390, −10.72939561331589434348042113165, −9.840206662931415288171243525265, −9.121995014615361078020740243320, −7.11035485822547423035216476827, −6.48342452544459584141962787996, −5.39715041228975191499845385672, −3.51772457500561680796944087903, −2.45216433528160721114039281414,
0.51984156926935839780257379443, 2.00398661048899986271159925255, 4.13989658385966436021985424269, 5.63801720927504702578403853124, 6.19840651913240189866083186741, 7.83555462276943320032702965336, 8.938767766721422977996568475340, 9.701848062261642803942217926863, 10.85124435840173979766444264313, 12.45606843771186665779107258303
