| L(s) = 1 | + (0.326 − 1.37i)2-s + (−0.0281 + 2.99i)3-s + (−1.78 − 0.897i)4-s + (1.73 + 1.28i)5-s + (4.11 + 1.01i)6-s + (3.67 + 2.41i)7-s + (−1.81 + 2.16i)8-s + (−8.99 − 0.168i)9-s + (2.33 − 1.96i)10-s + (1.53 + 13.1i)11-s + (2.74 − 5.33i)12-s + (−2.24 + 7.51i)13-s + (4.52 − 4.27i)14-s + (−3.91 + 5.15i)15-s + (2.38 + 3.20i)16-s + (32.0 − 5.64i)17-s + ⋯ |
| L(s) = 1 | + (0.163 − 0.688i)2-s + (−0.00937 + 0.999i)3-s + (−0.446 − 0.224i)4-s + (0.346 + 0.257i)5-s + (0.686 + 0.169i)6-s + (0.525 + 0.345i)7-s + (−0.227 + 0.270i)8-s + (−0.999 − 0.0187i)9-s + (0.233 − 0.196i)10-s + (0.139 + 1.19i)11-s + (0.228 − 0.444i)12-s + (−0.173 + 0.577i)13-s + (0.323 − 0.305i)14-s + (−0.261 + 0.343i)15-s + (0.149 + 0.200i)16-s + (1.88 − 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44392 + 0.623954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44392 + 0.623954i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.326 + 1.37i)T \) |
| 3 | \( 1 + (0.0281 - 2.99i)T \) |
| good | 5 | \( 1 + (-1.73 - 1.28i)T + (7.17 + 23.9i)T^{2} \) |
| 7 | \( 1 + (-3.67 - 2.41i)T + (19.4 + 44.9i)T^{2} \) |
| 11 | \( 1 + (-1.53 - 13.1i)T + (-117. + 27.9i)T^{2} \) |
| 13 | \( 1 + (2.24 - 7.51i)T + (-141. - 92.8i)T^{2} \) |
| 17 | \( 1 + (-32.0 + 5.64i)T + (271. - 98.8i)T^{2} \) |
| 19 | \( 1 + (4.44 - 25.2i)T + (-339. - 123. i)T^{2} \) |
| 23 | \( 1 + (2.91 + 4.43i)T + (-209. + 485. i)T^{2} \) |
| 29 | \( 1 + (-7.70 - 7.26i)T + (48.8 + 839. i)T^{2} \) |
| 31 | \( 1 + (1.26 + 21.7i)T + (-954. + 111. i)T^{2} \) |
| 37 | \( 1 + (43.5 + 15.8i)T + (1.04e3 + 879. i)T^{2} \) |
| 41 | \( 1 + (3.35 + 14.1i)T + (-1.50e3 + 754. i)T^{2} \) |
| 43 | \( 1 + (-5.82 + 13.4i)T + (-1.26e3 - 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-91.8 - 5.34i)T + (2.19e3 + 256. i)T^{2} \) |
| 53 | \( 1 + (29.2 - 16.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-3.98 + 34.0i)T + (-3.38e3 - 802. i)T^{2} \) |
| 61 | \( 1 + (25.1 - 12.6i)T + (2.22e3 - 2.98e3i)T^{2} \) |
| 67 | \( 1 + (62.5 + 66.2i)T + (-261. + 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-21.0 - 25.1i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (39.6 + 33.2i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-117. - 27.8i)T + (5.57e3 + 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-2.39 + 10.0i)T + (-6.15e3 - 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-85.7 + 102. i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-56.0 - 75.3i)T + (-2.69e3 + 9.01e3i)T^{2} \) |
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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
−12.26240212333551050659380409087, −11.94192402723243276367188658404, −10.49629079421830015744427998913, −10.00325118939323622732420487215, −9.060987486572342076590346358035, −7.75076336938047870586536594791, −5.91122865989110473321558757676, −4.85740394685129916583530799584, −3.69065471905353465999147877666, −2.09833549274603264136529858661,
1.03918307165922010819462590687, 3.21836044366142509943133480177, 5.20860187205088687390825630570, 6.02130752962667915115408995224, 7.30229225547843938795342254827, 8.114189536651798780678508403924, 9.045218202302183467264545893434, 10.58307013788132067428815813662, 11.74311575166041454305555112643, 12.71251670459470922200791821603
