L(s) = 1 | + (−1.28 + 0.596i)2-s + (−0.0980 + 0.995i)3-s + (1.28 − 1.53i)4-s + (−0.434 + 0.232i)5-s + (−0.468 − 1.33i)6-s + (−0.519 − 2.60i)7-s + (−0.736 + 2.73i)8-s + (−0.980 − 0.195i)9-s + (0.418 − 0.556i)10-s + (0.916 − 1.11i)11-s + (1.39 + 1.43i)12-s + (1.88 − 3.52i)13-s + (2.22 + 3.03i)14-s + (−0.188 − 0.455i)15-s + (−0.685 − 3.94i)16-s + (0.865 − 2.08i)17-s + ⋯ |
L(s) = 1 | + (−0.906 + 0.422i)2-s + (−0.0565 + 0.574i)3-s + (0.643 − 0.765i)4-s + (−0.194 + 0.103i)5-s + (−0.191 − 0.544i)6-s + (−0.196 − 0.986i)7-s + (−0.260 + 0.965i)8-s + (−0.326 − 0.0650i)9-s + (0.132 − 0.176i)10-s + (0.276 − 0.336i)11-s + (0.403 + 0.413i)12-s + (0.523 − 0.978i)13-s + (0.594 + 0.811i)14-s + (−0.0486 − 0.117i)15-s + (−0.171 − 0.985i)16-s + (0.209 − 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.755421 - 0.166912i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755421 - 0.166912i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 - 0.596i)T \) |
| 3 | \( 1 + (0.0980 - 0.995i)T \) |
good | 5 | \( 1 + (0.434 - 0.232i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.519 + 2.60i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.916 + 1.11i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.88 + 3.52i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.865 + 2.08i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.820 + 2.70i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (1.61 - 1.07i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-6.40 + 5.25i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-4.65 + 4.65i)T - 31iT^{2} \) |
| 37 | \( 1 + (-6.97 - 2.11i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-5.57 - 8.34i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.656 + 6.66i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (7.80 + 3.23i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (1.88 + 1.55i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (1.29 + 2.42i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-5.00 - 0.493i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (5.17 + 0.509i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-1.15 + 0.228i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.75 - 8.80i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (0.912 - 0.378i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (6.02 - 1.82i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (15.5 + 10.3i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-4.43 + 4.43i)T - 97iT^{2} \) |
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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
−11.12630291609404563541773649883, −10.10315093847408792874729305303, −9.668448230789390332600238471132, −8.409579785249320425749232519446, −7.72619480135927290920591957674, −6.62817355015312076829037466034, −5.69165713937736073244514872997, −4.33957890217940560026576889829, −2.96263086723384467514307435195, −0.74937988433260736527084423607,
1.52597656498385246535700831215, 2.74163242579487972447511036475, 4.21264496508934714424460835066, 6.04440334994052476621697030726, 6.71611795800680902261089253494, 7.975359796526419754615772769855, 8.627271078380499759404541302943, 9.452928077616428427661547236539, 10.47676546602652009739189991146, 11.49019831090589820425975543633