| L(s) = 1 | + (3.27 + 10.0i)2-s + (20.2 − 14.7i)3-s + (−65.1 + 47.3i)4-s + (47.0 + 30.2i)5-s + (214. + 155. i)6-s − 69.7·7-s + (−416. − 302. i)8-s + (118. − 364. i)9-s + (−150. + 573. i)10-s + (−71.7 − 220. i)11-s + (−622. + 1.91e3i)12-s + (109. − 335. i)13-s + (−228. − 703. i)14-s + (1.39e3 − 80.0i)15-s + (889. − 2.73e3i)16-s + (−515. − 374. i)17-s + ⋯ |
| L(s) = 1 | + (0.579 + 1.78i)2-s + (1.29 − 0.943i)3-s + (−2.03 + 1.47i)4-s + (0.841 + 0.540i)5-s + (2.43 + 1.76i)6-s − 0.537·7-s + (−2.29 − 1.67i)8-s + (0.487 − 1.49i)9-s + (−0.476 + 1.81i)10-s + (−0.178 − 0.550i)11-s + (−1.24 + 3.83i)12-s + (0.179 − 0.551i)13-s + (−0.311 − 0.959i)14-s + (1.60 − 0.0918i)15-s + (0.869 − 2.67i)16-s + (−0.432 − 0.314i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.62884 + 1.94667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62884 + 1.94667i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-47.0 - 30.2i)T \) |
| good | 2 | \( 1 + (-3.27 - 10.0i)T + (-25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (-20.2 + 14.7i)T + (75.0 - 231. i)T^{2} \) |
| 7 | \( 1 + 69.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + (71.7 + 220. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-109. + 335. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (515. + 374. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.57e3 - 1.14e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (458. + 1.41e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-588. + 427. i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (7.22e3 + 5.24e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (2.13e3 - 6.56e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (4.47e3 - 1.37e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 3.19e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.38e4 + 1.00e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.57e4 - 1.14e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.15e4 - 3.54e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (4.17e3 + 1.28e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.49e4 - 1.08e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-1.81e4 + 1.31e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.11e4 - 6.50e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.33e4 + 9.70e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.61e4 - 3.35e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (3.40e4 + 1.04e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-2.78e4 + 2.02e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−16.65508364764302846282518657166, −15.30749750524364351632930079538, −14.27986633462873869510390581533, −13.59795136642076445691267070716, −12.83760252061606099857405416783, −9.450219906847631052083508504470, −8.179075229416589249612117521567, −7.06079705924478065394886197059, −5.88757973357870585238138863167, −3.18808865794408522391718776063,
2.06363829357344406491962522812, 3.54577790667616617575492880571, 4.98330045857196191018595332750, 9.074173505516223170545337777363, 9.534958181446406349467641900681, 10.70169769884113061360387889990, 12.53721363545063695861045783293, 13.60015574497592723866464479561, 14.31371495472128057233843039146, 15.81632392628338729327620210352
