L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−1.40 + 1.74i)5-s + 0.999·6-s + (2.51 + 0.806i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.344 + 2.20i)10-s + (2.39 − 4.14i)11-s + (0.866 − 0.499i)12-s + 3.17i·13-s + (2.58 − 0.561i)14-s + (−2.08 + 0.806i)15-s + (−0.5 − 0.866i)16-s + (−4.52 − 2.61i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.627 + 0.778i)5-s + 0.408·6-s + (0.952 + 0.304i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.108 + 0.698i)10-s + (0.721 − 1.24i)11-s + (0.249 − 0.144i)12-s + 0.879i·13-s + (0.691 − 0.150i)14-s + (−0.538 + 0.208i)15-s + (−0.125 − 0.216i)16-s + (−1.09 − 0.633i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86279 - 0.00384614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86279 - 0.00384614i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (1.40 - 1.74i)T \) |
| 7 | \( 1 + (-2.51 - 0.806i)T \) |
good | 11 | \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.17iT - 13T^{2} \) |
| 17 | \( 1 + (4.52 + 2.61i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.58 + 2.74i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.21 - 3.58i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 + (2.08 - 3.61i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.21 + 3.58i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + (-9.97 + 5.75i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.02 + 4.05i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.36 - 9.29i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.55 + 6.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.28 + 4.78i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.08 - 10.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.39iT - 83T^{2} \) |
| 89 | \( 1 + (-4.78 - 8.28i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.27iT - 97T^{2} \) |
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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
−12.12082421852596853001762184593, −11.25793451842435427820500876404, −10.96496586666614498732694749128, −9.377394120960814994590564409264, −8.476260916646274556771374148427, −7.25629694209065545484842718082, −6.08981844504480088910461612731, −4.55545557001232425579490920050, −3.63262841114390890811497598370, −2.23008101522047363477640887926,
1.90260332856618686789969354837, 4.01173288424606710182633207537, 4.60925263765311503814384853582, 6.15699105848992261103530320961, 7.55032379730389619356343724990, 8.071192513024162168670154516520, 9.120103221756621446152879563053, 10.56237300770255017828927738752, 11.78200676369638946214251744298, 12.49889415297053287888749894311