| L(s) = 1 | + (0.732 − 2.73i)2-s + (7.29 + 4.21i)3-s + (−6.92 − 4i)4-s + (−12.4 − 46.3i)5-s + (16.8 − 16.8i)6-s + (−18.9 + 32.8i)7-s + (−16 + 15.9i)8-s + (−4.97 − 8.61i)9-s − 135.·10-s − 114. i·11-s + (−33.7 − 58.3i)12-s + (−5.81 − 21.6i)13-s + (75.8 + 75.8i)14-s + (104. − 391. i)15-s + (31.9 + 55.4i)16-s + (253. + 67.9i)17-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (0.811 + 0.468i)3-s + (−0.433 − 0.250i)4-s + (−0.497 − 1.85i)5-s + (0.468 − 0.468i)6-s + (−0.386 + 0.669i)7-s + (−0.250 + 0.249i)8-s + (−0.0614 − 0.106i)9-s − 1.35·10-s − 0.950i·11-s + (−0.234 − 0.405i)12-s + (−0.0344 − 0.128i)13-s + (0.386 + 0.386i)14-s + (0.465 − 1.73i)15-s + (0.124 + 0.216i)16-s + (0.877 + 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.693738 - 1.54501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.693738 - 1.54501i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.732 + 2.73i)T \) |
| 37 | \( 1 + (-1.20e3 - 652. i)T \) |
| good | 3 | \( 1 + (-7.29 - 4.21i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (12.4 + 46.3i)T + (-541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (18.9 - 32.8i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 114. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (5.81 + 21.6i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-253. - 67.9i)T + (7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (96.4 + 360. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-104. + 104. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-638. - 638. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-212. - 212. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-1.65e3 - 953. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.20e3 + 1.20e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 352.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (2.78e3 + 4.82e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.82e3 - 1.02e3i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (940. - 251. i)T + (1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-376. - 217. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (2.86e3 - 4.96e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 - 8.18e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (1.03e3 + 3.84e3i)T + (-3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (4.84e3 + 8.39e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (993. - 3.70e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-7.85e3 + 7.85e3i)T - 8.85e7iT^{2} \) |
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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
−13.17067567611774949428153797205, −12.44189095752617938689086290984, −11.46628565406431527980791592883, −9.726784188711332500998922588920, −8.833476899913424969867603876856, −8.333981167316287825573537171708, −5.65349066301884121465410604570, −4.36690292458541198609574898146, −3.03143702728574538664310122331, −0.78196671597674302101107859628,
2.66643095024490716397507454278, 3.93179644694411538912371405321, 6.27156416961099978078340588452, 7.42926379885621619582476991362, 7.78190606597937320635626735156, 9.724342937410539480507171945583, 10.75932675410790683682624557239, 12.23887486853263867077328993392, 13.61643267001163715361205948572, 14.38237375044811277504973742333
