L(s) = 1 | + (−1.20 − 0.694i)2-s + (3.08 − 4.18i)3-s + (−3.03 − 5.25i)4-s + (−2.5 + 4.33i)5-s + (−6.60 + 2.88i)6-s + (−11.1 − 14.8i)7-s + 19.5i·8-s + (−7.99 − 25.7i)9-s + (6.01 − 3.47i)10-s + (−11.5 + 6.66i)11-s + (−31.3 − 3.50i)12-s + 27.3i·13-s + (3.09 + 25.5i)14-s + (10.4 + 23.8i)15-s + (−10.7 + 18.5i)16-s + (−8.19 − 14.2i)17-s + ⋯ |
L(s) = 1 | + (−0.425 − 0.245i)2-s + (0.593 − 0.805i)3-s + (−0.379 − 0.657i)4-s + (−0.223 + 0.387i)5-s + (−0.449 + 0.196i)6-s + (−0.600 − 0.799i)7-s + 0.863i·8-s + (−0.296 − 0.955i)9-s + (0.190 − 0.109i)10-s + (−0.316 + 0.182i)11-s + (−0.754 − 0.0844i)12-s + 0.583i·13-s + (0.0590 + 0.487i)14-s + (0.179 + 0.409i)15-s + (−0.167 + 0.290i)16-s + (−0.116 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0413725 + 0.702404i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0413725 + 0.702404i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.08 + 4.18i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (11.1 + 14.8i)T \) |
good | 2 | \( 1 + (1.20 + 0.694i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (11.5 - 6.66i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 27.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (8.19 + 14.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-3.26 - 1.88i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (171. + 98.9i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 68.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (4.82 - 2.78i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-125. + 217. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 15.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-199. + 345. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-557. + 321. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (254. + 440. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-315. - 182. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (528. + 914. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 761. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-399. + 230. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (392. - 680. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-111. + 192. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.51e3iT - 9.12e5T^{2} \) |
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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
−12.88840672657581404778885262301, −11.63945449630741238997542931132, −10.40153849539123250617872449315, −9.547588129232248002379150196358, −8.365326464153328890749911064161, −7.21293927853096512119792142753, −6.09144628906937390295129261093, −4.06117198296828457351379462389, −2.23246532749687742467679166979, −0.40949830193974173037980147791,
2.94008497633848623204907536680, 4.17268704574299827643352375878, 5.70537906749537296877409449194, 7.64818534199456759214184309719, 8.500667307815603982586852869084, 9.325849922005961734368813748174, 10.23585805164046305859603274616, 11.82718853215556876739583217949, 12.87338674396737534957455007109, 13.73729875155397438049572645031