| L(s) = 1 | + (0.676 + 1.88i)2-s + (−2.21 − 2.02i)3-s + (−3.08 + 2.54i)4-s + (4.28 + 7.41i)5-s + (2.31 − 5.53i)6-s + (−3.75 + 6.50i)7-s + (−6.87 − 4.08i)8-s + (0.811 + 8.96i)9-s + (−11.0 + 13.0i)10-s + (4.74 − 8.22i)11-s + (11.9 + 0.605i)12-s + (9.54 − 5.51i)13-s + (−14.7 − 2.67i)14-s + (5.52 − 25.0i)15-s + (3.04 − 15.7i)16-s − 11.3i·17-s + ⋯ |
| L(s) = 1 | + (0.338 + 0.941i)2-s + (−0.738 − 0.674i)3-s + (−0.771 + 0.636i)4-s + (0.856 + 1.48i)5-s + (0.385 − 0.922i)6-s + (−0.536 + 0.929i)7-s + (−0.859 − 0.510i)8-s + (0.0901 + 0.995i)9-s + (−1.10 + 1.30i)10-s + (0.431 − 0.747i)11-s + (0.998 + 0.0504i)12-s + (0.734 − 0.424i)13-s + (−1.05 − 0.190i)14-s + (0.368 − 1.67i)15-s + (0.190 − 0.981i)16-s − 0.667i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.618840 + 0.961221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.618840 + 0.961221i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.676 - 1.88i)T \) |
| 3 | \( 1 + (2.21 + 2.02i)T \) |
| good | 5 | \( 1 + (-4.28 - 7.41i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (3.75 - 6.50i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.74 + 8.22i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-9.54 + 5.51i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11.3iT - 289T^{2} \) |
| 19 | \( 1 - 18.3iT - 361T^{2} \) |
| 23 | \( 1 + (-22.8 + 13.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (3.48 - 6.03i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-6.42 - 11.1i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 5.89iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-32.7 + 18.8i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (21.1 + 12.2i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (15.8 + 9.17i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 58.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-13.1 - 22.8i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (56.0 + 32.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (28.4 - 16.4i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 84.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 94.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (12.3 - 21.3i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-57.7 + 100. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-94.1 + 163. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.60466318358301696447240633645, −13.77390322566240959781294190941, −12.83208630220755116160797811986, −11.58500476306896850037685135127, −10.32183593341037336092229483964, −8.807121197531982029410637088513, −7.19582927439735168781277241090, −6.22108883296204714415740848049, −5.66900957801668760677892077919, −3.00574109095165419071628953622,
1.12690556190458894789199462667, 4.03625570020508130782246621037, 4.98683308339573502270194783333, 6.31200659739263601267413881613, 9.023533474710769377762691623598, 9.624911428645587020183550831031, 10.67480603669632677556147475717, 11.87193719314549066018826563397, 12.98243560588753214498463902309, 13.50981196367761285622111113965
