L(s) = 1 | + (2.68 + 4.64i)2-s + (−1.5 + 2.59i)3-s + (−10.3 + 17.9i)4-s + (2.5 + 4.33i)5-s − 16.0·6-s + (17.9 + 4.42i)7-s − 68.2·8-s + (−4.5 − 7.79i)9-s + (−13.4 + 23.2i)10-s + (27.8 − 48.2i)11-s + (−31.1 − 53.8i)12-s + 8.04·13-s + (27.6 + 95.3i)14-s − 15.0·15-s + (−100. − 173. i)16-s + (−48.9 + 84.7i)17-s + ⋯ |
L(s) = 1 | + (0.947 + 1.64i)2-s + (−0.288 + 0.499i)3-s + (−1.29 + 2.24i)4-s + (0.223 + 0.387i)5-s − 1.09·6-s + (0.970 + 0.239i)7-s − 3.01·8-s + (−0.166 − 0.288i)9-s + (−0.423 + 0.734i)10-s + (0.764 − 1.32i)11-s + (−0.748 − 1.29i)12-s + 0.171·13-s + (0.527 + 1.82i)14-s − 0.258·15-s + (−1.56 − 2.70i)16-s + (−0.697 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.249215 - 2.27003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.249215 - 2.27003i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 2.59i)T \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
| 7 | \( 1 + (-17.9 - 4.42i)T \) |
good | 2 | \( 1 + (-2.68 - 4.64i)T + (-4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-27.8 + 48.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 8.04T + 2.19e3T^{2} \) |
| 17 | \( 1 + (48.9 - 84.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (31.5 + 54.6i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-69.4 - 120. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-10.7 + 18.6i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (27.9 + 48.4i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 256.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-88.4 - 153. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-360. + 623. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (286. - 495. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (155. + 269. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-109. + 189. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 464.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-398. + 690. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (349. + 605. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 963.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (25.7 + 44.5i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−14.05207769324040988066201982787, −13.34813966217162846301514715696, −11.90522672188365442200159182240, −10.92055952845634577144279095810, −8.936941733447082869004626289708, −8.248965101409020136303742617005, −6.71984151644839542168956743082, −5.88619903536973070544774262361, −4.77515867893000584875773763433, −3.53760449866986078181581866039,
1.12925712937965342921742587316, 2.30499947698394323297720048032, 4.35543215036150837976215809969, 5.05554832144558460110712981789, 6.71780777874296692531497827706, 8.700426537763336421274441280148, 9.947990775814271618729366409893, 10.92959864963833406038121250027, 11.95975936076476193193439838928, 12.40031766410042730488865661110