| L(s) = 1 | + (4.13 + 3.15i)3-s + 1.18i·5-s + 7i·7-s + (7.12 + 26.0i)9-s − 64.7·11-s − 65.6·13-s + (−3.73 + 4.88i)15-s + 19.9i·17-s − 67.2i·19-s + (−22.0 + 28.9i)21-s − 79.6·23-s + 123.·25-s + (−52.7 + 130. i)27-s − 100. i·29-s + 278. i·31-s + ⋯ |
| L(s) = 1 | + (0.794 + 0.606i)3-s + 0.105i·5-s + 0.377i·7-s + (0.263 + 0.964i)9-s − 1.77·11-s − 1.39·13-s + (−0.0642 + 0.0841i)15-s + 0.285i·17-s − 0.812i·19-s + (−0.229 + 0.300i)21-s − 0.722·23-s + 0.988·25-s + (−0.375 + 0.926i)27-s − 0.643i·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9187323114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9187323114\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.13 - 3.15i)T \) |
| 7 | \( 1 - 7iT \) |
| good | 5 | \( 1 - 1.18iT - 125T^{2} \) |
| 11 | \( 1 + 64.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 67.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 79.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 278. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 45.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 12.4iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 368. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 639. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 537.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 533. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 348.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 517. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.76e3T + 9.12e5T^{2} \) |
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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
−11.42289566283809364876145919889, −10.32580075241132810575802546010, −9.897199622252661380790828805139, −8.702019958246354483191260024595, −7.959104242826633478743554395088, −7.00716453846120430831755274924, −5.31678205351631315309877044774, −4.68103962870355013246585482752, −3.04641341985562412549372688941, −2.30769328852370887801806578062,
0.26551291413410351865367436784, 2.08648006611851400862516846519, 3.07177688609574390650123567252, 4.55813731579284542714824947366, 5.75688815262567996099722589961, 7.23228144332946708046264857670, 7.68508672894417431633911830407, 8.635059681298025710002696375216, 9.830589506710217658916708926122, 10.42467879723238389736261995020
