| L(s) = 1 | + (−1.94 − 0.455i)2-s + (−0.811 − 1.40i)3-s + (3.58 + 1.77i)4-s + 0.300i·5-s + (0.939 + 3.10i)6-s + (5.76 + 3.32i)7-s + (−6.17 − 5.08i)8-s + (3.18 − 5.51i)9-s + (0.136 − 0.584i)10-s + (−2.55 − 4.43i)11-s + (−0.414 − 6.47i)12-s + (2.90 − 12.6i)13-s + (−9.70 − 9.10i)14-s + (0.421 − 0.243i)15-s + (9.70 + 12.7i)16-s + (11.9 − 20.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.973 − 0.227i)2-s + (−0.270 − 0.468i)3-s + (0.896 + 0.443i)4-s + 0.0600i·5-s + (0.156 + 0.517i)6-s + (0.823 + 0.475i)7-s + (−0.771 − 0.636i)8-s + (0.353 − 0.612i)9-s + (0.0136 − 0.0584i)10-s + (−0.232 − 0.402i)11-s + (−0.0345 − 0.539i)12-s + (0.223 − 0.974i)13-s + (−0.693 − 0.650i)14-s + (0.0281 − 0.0162i)15-s + (0.606 + 0.795i)16-s + (0.705 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.731516 - 0.484791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.731516 - 0.484791i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.94 + 0.455i)T \) |
| 13 | \( 1 + (-2.90 + 12.6i)T \) |
| good | 3 | \( 1 + (0.811 + 1.40i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 - 0.300iT - 25T^{2} \) |
| 7 | \( 1 + (-5.76 - 3.32i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (2.55 + 4.43i)T + (-60.5 + 104. i)T^{2} \) |
| 17 | \( 1 + (-11.9 + 20.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-4.87 + 8.44i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (7.86 - 4.53i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (4.28 - 2.47i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 45.9iT - 961T^{2} \) |
| 37 | \( 1 + (9.24 - 5.33i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-32.3 - 56.0i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-7.08 + 12.2i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + 11.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 67.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (34.9 - 60.4i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-27.6 - 15.9i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.0 - 38.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (15.0 + 8.68i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + 57.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 58.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + (5.70 + 9.88i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (17.8 - 30.9i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.00928781728375528590569885652, −12.03554538128338236023984436526, −11.28546053876630730817112140062, −10.14052672662972413461269508244, −8.943230979168116729121778330119, −7.896341233609440148672865908354, −6.85966161710612863504436973314, −5.42177746931824833195217743054, −3.01891545346603660268441221631, −1.05289041659480487132903089860,
1.74258801821502476088824455993, 4.34051354336033663336979339031, 5.80517463298812305353351896718, 7.34350754371251486743868935152, 8.192162664598746461635205016601, 9.534833955574771522090794980760, 10.53873239951670349116879603782, 11.16671707032073962516187149661, 12.44334395575684402332011901080, 14.04272085383676088135442822441
