L(s) = 1 | − 2.82·5-s + (−1.73 − 2i)7-s − 4.89·11-s − 6·13-s − 4.89i·17-s − 8.48i·23-s + 3.00·25-s − 3.46·31-s + (4.89 + 5.65i)35-s + 6.92i·37-s − 4.89i·41-s + 3.46·43-s − 9.79·47-s + (−1.00 + 6.92i)49-s − 9.79i·53-s + ⋯ |
L(s) = 1 | − 1.26·5-s + (−0.654 − 0.755i)7-s − 1.47·11-s − 1.66·13-s − 1.18i·17-s − 1.76i·23-s + 0.600·25-s − 0.622·31-s + (0.828 + 0.956i)35-s + 1.13i·37-s − 0.765i·41-s + 0.528·43-s − 1.42·47-s + (−0.142 + 0.989i)49-s − 1.34i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08719971000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08719971000\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.73 + 2i)T \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 4.89iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.180962266267512556514514868850, −7.915502288216763978932035798361, −7.02510816263117690613651209934, −6.79497573168893190359718002586, −5.29857144440827683120196535119, −4.79548728387719844137282401048, −4.02909142620585312349259260432, −3.01769987649798842576989762904, −2.45952938677274292317817405377, −0.45504247949776373646925496920,
0.05397370466575565446371796296, 1.98039850215446427567845861289, 2.91987525360664458311832583820, 3.58819991992635187360956304113, 4.57934262334097896801649402797, 5.34462423185125449871113726223, 5.98992446188356989555509363978, 7.12766023257852685535076376524, 7.72172083298000920221090032282, 8.033784298847169403109281172008