| L(s) = 1 | + (−0.459 − 1.76i)2-s + (−0.169 − 0.985i)3-s + (−1.16 + 0.648i)4-s + (0.417 − 1.93i)5-s + (−1.66 + 0.751i)6-s + (0.116 + 2.73i)7-s + (−0.846 − 0.883i)8-s + (−0.942 + 0.333i)9-s + (−3.61 + 0.153i)10-s + (1.00 − 1.80i)11-s + (0.835 + 1.03i)12-s + (−4.80 − 3.55i)13-s + (4.76 − 1.46i)14-s + (−1.97 − 0.0840i)15-s + (−2.56 + 4.16i)16-s + (−4.21 + 0.722i)17-s + ⋯ |
| L(s) = 1 | + (−0.325 − 1.24i)2-s + (−0.0975 − 0.569i)3-s + (−0.580 + 0.324i)4-s + (0.186 − 0.866i)5-s + (−0.678 + 0.306i)6-s + (0.0438 + 1.03i)7-s + (−0.299 − 0.312i)8-s + (−0.314 + 0.111i)9-s + (−1.14 + 0.0485i)10-s + (0.304 − 0.544i)11-s + (0.241 + 0.298i)12-s + (−1.33 − 0.985i)13-s + (1.27 − 0.390i)14-s + (−0.511 − 0.0217i)15-s + (−0.641 + 1.04i)16-s + (−1.02 + 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.285137 + 0.788826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.285137 + 0.788826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.169 + 0.985i)T \) |
| 149 | \( 1 + (12.1 - 0.601i)T \) |
| good | 2 | \( 1 + (0.459 + 1.76i)T + (-1.74 + 0.975i)T^{2} \) |
| 5 | \( 1 + (-0.417 + 1.93i)T + (-4.55 - 2.05i)T^{2} \) |
| 7 | \( 1 + (-0.116 - 2.73i)T + (-6.97 + 0.593i)T^{2} \) |
| 11 | \( 1 + (-1.00 + 1.80i)T + (-5.76 - 9.36i)T^{2} \) |
| 13 | \( 1 + (4.80 + 3.55i)T + (3.80 + 12.4i)T^{2} \) |
| 17 | \( 1 + (4.21 - 0.722i)T + (16.0 - 5.66i)T^{2} \) |
| 19 | \( 1 + (2.17 + 1.46i)T + (7.08 + 17.6i)T^{2} \) |
| 23 | \( 1 + (-4.32 + 3.19i)T + (6.73 - 21.9i)T^{2} \) |
| 29 | \( 1 + (0.656 + 0.746i)T + (-3.68 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.94 + 2.62i)T + (-9.07 + 29.6i)T^{2} \) |
| 37 | \( 1 + (-5.02 - 2.80i)T + (19.3 + 31.5i)T^{2} \) |
| 41 | \( 1 + (-0.933 - 0.574i)T + (18.4 + 36.6i)T^{2} \) |
| 43 | \( 1 + (-2.15 + 0.276i)T + (41.6 - 10.8i)T^{2} \) |
| 47 | \( 1 + (-0.647 + 0.437i)T + (17.5 - 43.6i)T^{2} \) |
| 53 | \( 1 + (-1.13 + 8.88i)T + (-51.2 - 13.3i)T^{2} \) |
| 59 | \( 1 + (5.97 - 2.40i)T + (42.5 - 40.8i)T^{2} \) |
| 61 | \( 1 + (3.41 - 0.889i)T + (53.2 - 29.7i)T^{2} \) |
| 67 | \( 1 + (-2.21 + 7.22i)T + (-55.5 - 37.5i)T^{2} \) |
| 71 | \( 1 + (-15.7 - 3.39i)T + (64.6 + 29.2i)T^{2} \) |
| 73 | \( 1 + (-2.98 - 0.254i)T + (71.9 + 12.3i)T^{2} \) |
| 79 | \( 1 + (0.422 + 4.96i)T + (-77.8 + 13.3i)T^{2} \) |
| 83 | \( 1 + (-0.705 + 8.29i)T + (-81.8 - 14.0i)T^{2} \) |
| 89 | \( 1 + (4.76 - 2.40i)T + (52.9 - 71.5i)T^{2} \) |
| 97 | \( 1 + (8.94 + 1.14i)T + (93.8 + 24.4i)T^{2} \) |
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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
−10.78358233940272577340023064797, −9.591070142566281287801624635915, −8.950529875982197559534286305384, −8.232163701006987487406775438268, −6.77018474202823971420383592109, −5.69953341235764887226716267289, −4.64016588610207030403124298247, −2.91066205659963573586576080494, −2.08963527430911755612662854683, −0.56134018516280254400452222193,
2.51317525954870553699728368722, 4.13529122581125482637416357041, 5.05784321121294913770986680473, 6.45308125606121376504707713616, 7.00881376853515289831623838930, 7.62057775697714458859073535039, 9.024907976742327490043546982638, 9.616225397472717320824113716647, 10.67394671200780562838550805524, 11.33509389142592078725488346246
