L(s) = 1 | + (1.5 + 2.59i)3-s + (−0.0642 + 0.111i)5-s + (0.866 − 18.4i)7-s + (−4.5 + 7.79i)9-s + (27.0 + 46.7i)11-s − 50.2·13-s − 0.385·15-s + (65.7 + 113. i)17-s + (45.7 − 79.2i)19-s + (49.3 − 25.4i)21-s + (89.7 − 155. i)23-s + (62.4 + 108. i)25-s − 27·27-s − 69.8·29-s + (163. + 283. i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.00575 + 0.00996i)5-s + (0.0467 − 0.998i)7-s + (−0.166 + 0.288i)9-s + (0.740 + 1.28i)11-s − 1.07·13-s − 0.00664·15-s + (0.937 + 1.62i)17-s + (0.552 − 0.956i)19-s + (0.512 − 0.264i)21-s + (0.813 − 1.40i)23-s + (0.499 + 0.865i)25-s − 0.192·27-s − 0.447·29-s + (0.946 + 1.63i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.025406355\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025406355\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (-0.866 + 18.4i)T \) |
good | 5 | \( 1 + (0.0642 - 0.111i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-27.0 - 46.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 50.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-65.7 - 113. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-45.7 + 79.2i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-89.7 + 155. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 69.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-163. - 283. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (150. - 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 296.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (180. - 311. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (0.917 + 1.58i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (26.6 + 46.1i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-54.0 + 93.6i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-421. - 729. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 241.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-103. - 179. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-279. + 484. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 986.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-221. + 383. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 740.T + 9.12e5T^{2} \) |
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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
−11.06902197303340073151544809636, −10.23084981457941925747122610913, −9.624518599272520102165114710146, −8.528294152787050997554307563811, −7.37505220604542685389933494302, −6.70153255079097296054252728568, −4.99141411179762815320219034927, −4.28134194355313791893139049414, −3.01176139457744335426120800341, −1.32599867473000601955002908531,
0.794934256399545305208804907983, 2.44533707818752122913459930794, 3.46618836089878856741497388621, 5.22600964195533709368425776442, 5.98674079534742615482379063908, 7.27184372070020754936218935828, 8.070527783082231358381892762878, 9.204376680301887323530008688352, 9.689663101646931779734534805799, 11.33092736841616967924999644320