L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s − 4·13-s + (−2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (0.499 − 0.866i)18-s + (−1 − 1.73i)19-s + (−2 − 1.73i)21-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.144 − 0.249i)12-s − 1.10·13-s + (−0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.117 − 0.204i)18-s + (−0.229 − 0.397i)19-s + (−0.436 − 0.377i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6230073291\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6230073291\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−10.21335969402079982410963072962, −9.614504355951646771465534462245, −8.667591786671410119793608116262, −8.060266459254038073469392377787, −6.78683655162858056919488096929, −6.22854308036208720534033800373, −5.17594272644048480983069951448, −4.64066406847561342282695939533, −3.39627365683952838695544182477, −2.28816009574158558705797445878,
0.24439565688008791105254155447, 1.70702419671036469922504204819, 2.89938881670955343353733750697, 4.05280261251976099551174012308, 4.92021958880353933185533020363, 5.88724220732158892434040578994, 6.97629222482906056024192964112, 7.48187883509705534468151499347, 8.618310615937438547284024461429, 9.730538780379656371421190911888