| L(s) = 1 | + (1.91 − 2.27i)3-s + (−16.1 + 5.87i)5-s + (23.9 + 41.5i)7-s + (12.5 + 71.0i)9-s + (41.5 − 72.0i)11-s + (175. + 209. i)13-s + (−17.4 + 47.9i)15-s + (17.7 − 100. i)17-s + (−229. + 278. i)19-s + (140. + 24.7i)21-s + (806. + 293. i)23-s + (−252. + 212. i)25-s + (394. + 227. i)27-s + (−753. + 132. i)29-s + (−222. + 128. i)31-s + ⋯ |
| L(s) = 1 | + (0.212 − 0.252i)3-s + (−0.645 + 0.235i)5-s + (0.489 + 0.847i)7-s + (0.154 + 0.877i)9-s + (0.343 − 0.595i)11-s + (1.03 + 1.23i)13-s + (−0.0776 + 0.213i)15-s + (0.0615 − 0.349i)17-s + (−0.636 + 0.771i)19-s + (0.318 + 0.0561i)21-s + (1.52 + 0.555i)23-s + (−0.404 + 0.339i)25-s + (0.540 + 0.312i)27-s + (−0.896 + 0.158i)29-s + (−0.231 + 0.133i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.40038 + 0.794288i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40038 + 0.794288i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (229. - 278. i)T \) |
| good | 3 | \( 1 + (-1.91 + 2.27i)T + (-14.0 - 79.7i)T^{2} \) |
| 5 | \( 1 + (16.1 - 5.87i)T + (478. - 401. i)T^{2} \) |
| 7 | \( 1 + (-23.9 - 41.5i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-41.5 + 72.0i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-175. - 209. i)T + (-4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-17.7 + 100. i)T + (-7.84e4 - 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-806. - 293. i)T + (2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (753. - 132. i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (222. - 128. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.60e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-531. + 633. i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (2.42e3 - 881. i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (100. + 569. i)T + (-4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 + (-1.02e3 + 2.81e3i)T + (-6.04e6 - 5.07e6i)T^{2} \) |
| 59 | \( 1 + (-1.56e3 - 275. i)T + (1.13e7 + 4.14e6i)T^{2} \) |
| 61 | \( 1 + (2.68e3 + 977. i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-3.11e3 + 549. i)T + (1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-2.71e3 - 7.44e3i)T + (-1.94e7 + 1.63e7i)T^{2} \) |
| 73 | \( 1 + (-4.10e3 - 3.44e3i)T + (4.93e6 + 2.79e7i)T^{2} \) |
| 79 | \( 1 + (-3.25e3 + 3.87e3i)T + (-6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-3.64e3 - 6.31e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.89e3 + 3.45e3i)T + (-1.08e7 + 6.17e7i)T^{2} \) |
| 97 | \( 1 + (1.04e3 + 183. i)T + (8.31e7 + 3.02e7i)T^{2} \) |
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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
−13.99462191615067012142538936081, −12.91181595798043705813153546015, −11.51646014828585042702393260247, −11.03470410203533250218800303379, −9.132195570382000143629149758469, −8.251086351252201592731168552563, −7.00836361968564673044440754001, −5.44575593508301861353910252016, −3.75716184302095309797458506576, −1.85240448348846423077152042093,
0.890234180879373337428538696814, 3.50223139696215666476448643058, 4.63086229611577996359061707413, 6.54825719309383896871597501181, 7.86728632832099533441417948722, 8.930893378487488877237291377487, 10.32865562607588685276192980641, 11.29810049333177928899067852782, 12.56267365498870187288103810674, 13.51298095305995369624505142537
