| L(s) = 1 | + (1.40 − 0.123i)2-s + (−0.0231 − 0.0276i)3-s + (1.96 − 0.347i)4-s + (0.534 + 1.14i)5-s + (−0.0360 − 0.0360i)6-s + (8.81 + 3.20i)7-s + (2.73 − 0.732i)8-s + (1.56 − 8.86i)9-s + (0.893 + 1.54i)10-s + (−13.9 − 8.05i)11-s + (−0.0552 − 0.0463i)12-s + (−14.7 + 21.1i)13-s + (12.8 + 3.43i)14-s + (0.0192 − 0.0412i)15-s + (3.75 − 1.36i)16-s + (9.52 − 6.67i)17-s + ⋯ |
| L(s) = 1 | + (0.704 − 0.0616i)2-s + (−0.00772 − 0.00920i)3-s + (0.492 − 0.0868i)4-s + (0.106 + 0.229i)5-s + (−0.00600 − 0.00600i)6-s + (1.25 + 0.458i)7-s + (0.341 − 0.0915i)8-s + (0.173 − 0.984i)9-s + (0.0893 + 0.154i)10-s + (−1.26 − 0.732i)11-s + (−0.00460 − 0.00386i)12-s + (−1.13 + 1.62i)13-s + (0.915 + 0.245i)14-s + (0.00128 − 0.00275i)15-s + (0.234 − 0.0855i)16-s + (0.560 − 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.86542 - 0.0307998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86542 - 0.0307998i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.123i)T \) |
| 37 | \( 1 + (-20.6 - 30.6i)T \) |
| good | 3 | \( 1 + (0.0231 + 0.0276i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (-0.534 - 1.14i)T + (-16.0 + 19.1i)T^{2} \) |
| 7 | \( 1 + (-8.81 - 3.20i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (13.9 + 8.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (14.7 - 21.1i)T + (-57.8 - 158. i)T^{2} \) |
| 17 | \( 1 + (-9.52 + 6.67i)T + (98.8 - 271. i)T^{2} \) |
| 19 | \( 1 + (26.5 + 2.32i)T + (355. + 62.6i)T^{2} \) |
| 23 | \( 1 + (7.09 - 26.4i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (8.63 + 32.2i)T + (-728. + 420.5i)T^{2} \) |
| 31 | \( 1 + (7.36 - 7.36i)T - 961iT^{2} \) |
| 41 | \( 1 + (-38.4 + 6.78i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (13.6 + 13.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (16.1 + 28.0i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-63.1 + 22.9i)T + (2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-61.7 - 28.7i)T + (2.23e3 + 2.66e3i)T^{2} \) |
| 61 | \( 1 + (-53.4 - 37.4i)T + (1.27e3 + 3.49e3i)T^{2} \) |
| 67 | \( 1 + (17.1 - 47.1i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (74.0 - 62.1i)T + (875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 - 51.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-1.38 - 2.96i)T + (-4.01e3 + 4.78e3i)T^{2} \) |
| 83 | \( 1 + (-6.01 + 34.0i)T + (-6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-33.6 + 72.1i)T + (-5.09e3 - 6.06e3i)T^{2} \) |
| 97 | \( 1 + (12.9 - 48.1i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−14.49509541329218883421362192810, −13.32148467600967581002851431152, −11.99770871530221163854079070273, −11.39091033008930598906923838125, −9.981191371439491818043624319513, −8.483570610399575612075934367466, −7.10002330176128127254034607179, −5.64971185135133032505352935875, −4.36129523718686959190265842166, −2.35164497935453821754583287554,
2.31641686629467325358686639733, 4.65944789262696164646263220195, 5.34931254363175286873638764565, 7.51200763253392993595798156257, 8.081602520470941386799219269492, 10.40089184031047674990980835923, 10.79079645977402787710330724079, 12.63942149284642229297224873732, 12.97966304434721058878800981272, 14.57103534724291681711874381118
