| L(s) = 1 | + (−0.308 − 0.674i)2-s + (1.68 − 0.494i)3-s + (4.87 − 5.63i)4-s + (3.36 + 2.16i)5-s + (−0.852 − 0.984i)6-s + (0.387 + 2.69i)7-s + (−10.9 − 3.22i)8-s + (−20.1 + 12.9i)9-s + (0.422 − 2.93i)10-s + (−15.2 + 33.3i)11-s + (5.43 − 11.8i)12-s + (−1.29 + 8.98i)13-s + (1.69 − 1.09i)14-s + (6.74 + 1.98i)15-s + (−7.27 − 50.5i)16-s + (3.08 + 3.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.108 − 0.238i)2-s + (0.324 − 0.0952i)3-s + (0.609 − 0.703i)4-s + (0.301 + 0.193i)5-s + (−0.0580 − 0.0669i)6-s + (0.0209 + 0.145i)7-s + (−0.485 − 0.142i)8-s + (−0.745 + 0.478i)9-s + (0.0133 − 0.0929i)10-s + (−0.416 + 0.912i)11-s + (0.130 − 0.286i)12-s + (−0.0275 + 0.191i)13-s + (0.0324 − 0.0208i)14-s + (0.116 + 0.0341i)15-s + (−0.113 − 0.790i)16-s + (0.0439 + 0.0507i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20644 - 0.332855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20644 - 0.332855i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-51.5 - 97.5i)T \) |
| good | 2 | \( 1 + (0.308 + 0.674i)T + (-5.23 + 6.04i)T^{2} \) |
| 3 | \( 1 + (-1.68 + 0.494i)T + (22.7 - 14.5i)T^{2} \) |
| 5 | \( 1 + (-3.36 - 2.16i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (-0.387 - 2.69i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (15.2 - 33.3i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (1.29 - 8.98i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (-3.08 - 3.55i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-55.6 + 64.2i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (25.6 + 29.6i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (83.4 + 24.5i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-123. + 79.6i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (297. + 191. i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-335. + 98.5i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 - 321.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (59.0 + 410. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (84.7 - 589. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (818. + 240. i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (-418. - 915. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (162. + 355. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (523. - 604. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (93.2 - 648. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-713. + 458. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-866. + 254. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-905. - 582. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−17.42118748606569408062659576635, −15.82246112010569874681641548369, −14.77137111478863612585577396916, −13.60014273288918137863842045661, −11.85166990404353308306590920679, −10.60177896282575909603851205814, −9.283803509431090043552704517039, −7.31251262688661845401140193421, −5.52270572792799566778017722099, −2.38299282443334420458724007699,
3.16115758589308165416386988525, 5.96475622063632928453370065328, 7.79436523196937955112952996986, 9.037656136884837026097588438034, 10.96548441388744922168580160711, 12.28881901770303990102858340434, 13.72658212642331148537281408480, 15.09081076613982145073266812303, 16.38609451565898571365619302693, 17.22454656718264017026882959943
