L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.73 + 2.99i)5-s + 0.999·6-s + (−2.36 + 1.17i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.73 + 2.99i)10-s + (−0.139 + 0.240i)11-s + (0.499 + 0.866i)12-s + 6.92·13-s + (−2.20 − 1.46i)14-s + 3.46·15-s + (−0.5 − 0.866i)16-s + (−2.89 + 5.02i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.773 + 1.34i)5-s + 0.408·6-s + (−0.895 + 0.444i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.547 + 0.947i)10-s + (−0.0419 + 0.0726i)11-s + (0.144 + 0.249i)12-s + 1.91·13-s + (−0.588 − 0.391i)14-s + 0.893·15-s + (−0.125 − 0.216i)16-s + (−0.703 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09002 + 1.63977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09002 + 1.63977i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.36 - 1.17i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.73 - 2.99i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.139 - 0.240i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.92T + 13T^{2} \) |
| 17 | \( 1 + (2.89 - 5.02i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.54 + 2.67i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.55T + 29T^{2} \) |
| 31 | \( 1 + (3.67 - 6.36i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.62 - 4.53i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + (-0.896 - 1.55i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.19 - 7.26i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.89 + 6.75i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.56 + 7.90i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.86 + 11.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.684T + 71T^{2} \) |
| 73 | \( 1 + (-0.962 + 1.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.703 + 1.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.146T + 83T^{2} \) |
| 89 | \( 1 + (-6.85 - 11.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
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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
−10.75142413347626710760792924457, −9.464096538150909497361150331944, −8.826136370180037233323332337709, −7.82146467849756246883187208172, −6.73852663617873862987013889889, −6.23493396949902640735277256124, −5.81475812307985706407112324399, −3.91437356586083488991105772987, −3.12544290773258517584730385880, −1.99926828918830973008556994687,
0.866527039500685051311370500646, 2.26770258926195477928654559980, 3.70616123147252652625851179420, 4.27657181975494744598299067629, 5.59255709685485896435733089681, 6.00296792309526896256002410284, 7.51929254140973380622172328325, 8.873879526067878586492409933958, 9.192600135354117176982325919063, 9.835541618795100104217648666737