| L(s) = 1 | + (−1.34 + 0.447i)2-s + (0.466 + 0.564i)3-s + (1.59 − 1.20i)4-s + (0.134 − 2.23i)5-s + (−0.879 − 0.547i)6-s + (0.385 − 0.125i)7-s + (−1.60 + 2.32i)8-s + (0.461 − 2.41i)9-s + (0.819 + 3.05i)10-s + (0.382 − 3.02i)11-s + (1.42 + 0.341i)12-s + (−0.145 + 0.763i)13-s + (−0.461 + 0.340i)14-s + (1.32 − 0.966i)15-s + (1.11 − 3.84i)16-s + (1.04 + 4.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.948 + 0.316i)2-s + (0.269 + 0.325i)3-s + (0.799 − 0.600i)4-s + (0.0600 − 0.998i)5-s + (−0.358 − 0.223i)6-s + (0.145 − 0.0473i)7-s + (−0.567 + 0.823i)8-s + (0.153 − 0.806i)9-s + (0.259 + 0.965i)10-s + (0.115 − 0.912i)11-s + (0.411 + 0.0985i)12-s + (−0.0404 + 0.211i)13-s + (−0.123 + 0.0910i)14-s + (0.341 − 0.249i)15-s + (0.278 − 0.960i)16-s + (0.254 + 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00754 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00754 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.664002 - 0.659012i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.664002 - 0.659012i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.34 - 0.447i)T \) |
| 5 | \( 1 + (-0.134 + 2.23i)T \) |
| good | 3 | \( 1 + (-0.466 - 0.564i)T + (-0.562 + 2.94i)T^{2} \) |
| 7 | \( 1 + (-0.385 + 0.125i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.382 + 3.02i)T + (-10.6 - 2.73i)T^{2} \) |
| 13 | \( 1 + (0.145 - 0.763i)T + (-12.0 - 4.78i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 4.08i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (3.60 + 2.98i)T + (3.56 + 18.6i)T^{2} \) |
| 23 | \( 1 + (-1.76 + 1.88i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-3.76 + 0.236i)T + (28.7 - 3.63i)T^{2} \) |
| 31 | \( 1 + (6.56 - 1.68i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (10.0 + 5.53i)T + (19.8 + 31.2i)T^{2} \) |
| 41 | \( 1 + (-2.61 + 2.45i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-7.99 + 5.81i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (1.41 + 3.58i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (5.77 - 9.10i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (-0.214 + 0.100i)T + (37.6 - 45.4i)T^{2} \) |
| 61 | \( 1 + (9.12 - 9.71i)T + (-3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.975 + 15.5i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (-12.7 + 5.04i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-4.65 - 2.18i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (4.55 + 5.50i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (5.94 - 7.19i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (-4.62 + 9.83i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-14.1 + 0.887i)T + (96.2 - 12.1i)T^{2} \) |
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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
−9.423718914932013691763819452083, −8.851174214502227357014709374991, −8.513892966188615568626580819492, −7.42328184893866064458159033341, −6.38851388256632015917596796265, −5.66895191075228402128104637952, −4.51326200771103776937554034154, −3.37751955509981252966081811917, −1.83243052539459744709834212428, −0.55719144378729571880862519032,
1.68572898480607080010868146188, 2.51250831496029997233166924464, 3.51874243840900641955646578588, 4.93990219115719770854117632566, 6.32431769133106347040707841785, 7.13502390338170994979673596873, 7.66099299526858417444204967344, 8.421881381047107060551097652587, 9.555436993484047139357949334502, 10.09585747482550437259162020360
