L(s) = 1 | + (0.795 − 2.18i)2-s + (−2.60 − 2.18i)4-s + (2.16 − 0.545i)5-s + (−0.124 − 0.0716i)7-s + (−2.82 + 1.63i)8-s + (0.531 − 5.17i)10-s + (−1.40 − 2.43i)11-s + (1.68 + 0.296i)13-s + (−0.255 + 0.214i)14-s + (0.135 + 0.766i)16-s + (1.21 − 3.34i)17-s + (−2.82 + 3.32i)19-s + (−6.84 − 3.32i)20-s + (−6.44 + 1.13i)22-s + (4.62 − 5.51i)23-s + ⋯ |
L(s) = 1 | + (0.562 − 1.54i)2-s + (−1.30 − 1.09i)4-s + (0.969 − 0.244i)5-s + (−0.0469 − 0.0270i)7-s + (−0.999 + 0.576i)8-s + (0.168 − 1.63i)10-s + (−0.424 − 0.735i)11-s + (0.466 + 0.0822i)13-s + (−0.0682 + 0.0572i)14-s + (0.0338 + 0.191i)16-s + (0.295 − 0.811i)17-s + (−0.647 + 0.762i)19-s + (−1.53 − 0.742i)20-s + (−1.37 + 0.242i)22-s + (0.964 − 1.14i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.223996 - 2.21074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.223996 - 2.21074i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.16 + 0.545i)T \) |
| 19 | \( 1 + (2.82 - 3.32i)T \) |
good | 2 | \( 1 + (-0.795 + 2.18i)T + (-1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (0.124 + 0.0716i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.40 + 2.43i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.68 - 0.296i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.21 + 3.34i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.62 + 5.51i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (7.11 - 2.58i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.42 - 4.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.49iT - 37T^{2} \) |
| 41 | \( 1 + (-0.0325 - 0.184i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.06 - 8.42i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.25 - 3.44i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.05 + 1.25i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-5.04 - 1.83i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.74 - 6.50i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.05 - 5.64i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.414 - 0.347i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (10.3 - 1.81i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.01 - 11.4i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.57 - 3.79i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.251 + 1.42i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.01 - 8.27i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.03025596773199493934867351311, −9.228637093176971159056723194804, −8.545347873087512887596515787465, −7.10020599296017995917247096031, −5.83668692694677369827939094899, −5.21857850401196721478708541771, −4.12291493601645407851274482503, −3.06194659187977376642343836924, −2.16651069195033608556963948232, −0.946303913280997771573578916922,
1.98366032494782283529100799260, 3.53690851524334100163886602608, 4.70768137182161938511449980987, 5.54299041914581998955441596625, 6.16276559568166782773033400913, 7.04002588482487013771568185904, 7.69706024499862499407078925782, 8.744157923189984328889177391341, 9.464170200213415081713373770104, 10.44529626586890354992714542499