L(s) = 1 | + (−2.14 + 3.71i)5-s + (1.40 + 2.24i)7-s + (−1.90 − 3.30i)11-s + 3.28·13-s + (−0.405 − 0.702i)17-s + (−3.54 + 6.14i)19-s + (−3.23 + 5.60i)23-s + (−6.69 − 11.5i)25-s − 3.81·29-s + (−1.64 − 2.84i)31-s + (−11.3 + 0.413i)35-s + (2.88 − 4.99i)37-s − 2.09·41-s − 8.76·43-s + (1.66 − 2.88i)47-s + ⋯ |
L(s) = 1 | + (−0.958 + 1.66i)5-s + (0.531 + 0.847i)7-s + (−0.574 − 0.995i)11-s + 0.911·13-s + (−0.0983 − 0.170i)17-s + (−0.814 + 1.41i)19-s + (−0.675 + 1.16i)23-s + (−1.33 − 2.31i)25-s − 0.707·29-s + (−0.295 − 0.511i)31-s + (−1.91 + 0.0698i)35-s + (0.473 − 0.820i)37-s − 0.327·41-s − 1.33·43-s + (0.243 − 0.421i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.176622 + 0.821015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.176622 + 0.821015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.40 - 2.24i)T \) |
good | 5 | \( 1 + (2.14 - 3.71i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.90 + 3.30i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 + (0.405 + 0.702i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.54 - 6.14i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.23 - 5.60i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 + (1.64 + 2.84i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.88 + 4.99i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.09T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 + (-1.66 + 2.88i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.93 - 8.54i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.73 - 3.01i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.97 - 5.15i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.76 - 3.05i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 + (-5.19 - 8.99i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.57 + 4.45i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.71T + 83T^{2} \) |
| 89 | \( 1 + (-6.26 + 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.04T + 97T^{2} \) |
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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
−10.81970813279401433812018032087, −10.15404267434762176090316176522, −8.781323927432560547920974152563, −8.025088639875266949740466667964, −7.44510822250563360814676744662, −6.17459397727687555999343201396, −5.70353570005274622091750501652, −3.97622422468823925535762190792, −3.31180128413584180940656848660, −2.14043023330103465195535916081,
0.42184164588472040191055514610, 1.79236073353234755384242433461, 3.75535235404111042530251109991, 4.60743350287344586503040100912, 5.01862145242924808445899959582, 6.58832155949694652548789441255, 7.60268739846125142553707236974, 8.317310902327643618157249926301, 8.852276224660831467290255083062, 9.970768820257272654255195734983