| L(s) = 1 | + (1.35 + 1.47i)2-s + (2.76 − 1.16i)3-s + (−0.340 + 3.98i)4-s + (0.0166 − 0.00958i)5-s + (5.45 + 2.50i)6-s + (−4.07 − 2.35i)7-s + (−6.33 + 4.88i)8-s + (6.29 − 6.42i)9-s + (0.0365 + 0.0114i)10-s + (2.84 − 4.93i)11-s + (3.68 + 11.4i)12-s + (−10.0 + 5.80i)13-s + (−2.04 − 9.17i)14-s + (0.0347 − 0.0458i)15-s + (−15.7 − 2.71i)16-s − 0.376·17-s + ⋯ |
| L(s) = 1 | + (0.676 + 0.736i)2-s + (0.921 − 0.387i)3-s + (−0.0852 + 0.996i)4-s + (0.00332 − 0.00191i)5-s + (0.908 + 0.417i)6-s + (−0.581 − 0.335i)7-s + (−0.791 + 0.611i)8-s + (0.699 − 0.714i)9-s + (0.00365 + 0.00114i)10-s + (0.259 − 0.448i)11-s + (0.307 + 0.951i)12-s + (−0.773 + 0.446i)13-s + (−0.146 − 0.655i)14-s + (0.00231 − 0.00305i)15-s + (−0.985 − 0.169i)16-s − 0.0221·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.86936 + 0.741229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86936 + 0.741229i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 - 1.47i)T \) |
| 3 | \( 1 + (-2.76 + 1.16i)T \) |
| good | 5 | \( 1 + (-0.0166 + 0.00958i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (4.07 + 2.35i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.84 + 4.93i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (10.0 - 5.80i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 0.376T + 289T^{2} \) |
| 19 | \( 1 + 15.0T + 361T^{2} \) |
| 23 | \( 1 + (-39.1 + 22.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (32.0 + 18.4i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-26.3 + 15.2i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 53.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-29.0 - 50.2i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (23.0 - 39.9i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-34.2 - 19.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 0.989iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-29.4 - 50.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-75.1 - 43.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (34.1 + 59.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 42.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 26.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (121. + 69.9i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-40.9 + 71.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 42.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-55.9 + 96.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.63695479176107794649204852012, −13.38769417647866947770830962088, −12.92972605463085752050961196186, −11.57054162096208421845194663923, −9.632341943444195793438987290126, −8.534481442306127321371647703041, −7.30791479250418476754567322862, −6.35916603678074223484967931594, −4.37683722820960323757506204685, −2.91622525532924149537080869332,
2.39570304099942746937590593756, 3.78912932850469046861325591329, 5.26048018784726102546873090955, 7.08793141275592653384853159919, 8.932908896162485930801495533980, 9.790254674905309744986009281634, 10.84112154962155748694274307948, 12.38038012407592052529704554251, 13.08707731226084021150212440866, 14.25884360822747025785920123117
