L(s) = 1 | + 1.73i·7-s + (−0.5 + 0.866i)13-s − 19-s + (0.5 − 0.866i)25-s + 1.73i·31-s − 37-s + (−1.5 + 0.866i)43-s − 1.99·49-s + (0.5 − 0.866i)61-s + (1.5 + 0.866i)67-s + (−0.5 − 0.866i)73-s + (1.5 − 0.866i)79-s + (−1.49 − 0.866i)91-s + (1 + 1.73i)97-s + 1.73i·103-s + ⋯ |
L(s) = 1 | + 1.73i·7-s + (−0.5 + 0.866i)13-s − 19-s + (0.5 − 0.866i)25-s + 1.73i·31-s − 37-s + (−1.5 + 0.866i)43-s − 1.99·49-s + (0.5 − 0.866i)61-s + (1.5 + 0.866i)67-s + (−0.5 − 0.866i)73-s + (1.5 − 0.866i)79-s + (−1.49 − 0.866i)91-s + (1 + 1.73i)97-s + 1.73i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9498717683\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9498717683\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - 1.73iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.956347155768019861844335080888, −8.717835917660286676892196492538, −7.899988869357729301837164397719, −6.61629568662586173017187274181, −6.42056799225751556685081445375, −5.19079921750167853075494526374, −4.81377805780149879784234951135, −3.52801838285196907328727236870, −2.52484782754430229248165302089, −1.81980078647027655032016534291,
0.58543185996533308917789117660, 1.94496749112244977968865216055, 3.25102498689477295501650107012, 3.97943034627320081865768966328, 4.77565610126035143931323831375, 5.65695881297572421585454867291, 6.73317677564875107917777856273, 7.22247672072157391668508463573, 7.952547732500360601660244119708, 8.652040467403177403114760016448