L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.707 + 0.707i)7-s + 0.999i·8-s + (−0.866 + 0.499i)10-s + (1.67 − 0.965i)11-s − 0.517i·13-s + (0.258 + 0.965i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)19-s − 0.999·20-s + 1.93·22-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.258 − 0.448i)26-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.707 + 0.707i)7-s + 0.999i·8-s + (−0.866 + 0.499i)10-s + (1.67 − 0.965i)11-s − 0.517i·13-s + (0.258 + 0.965i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)19-s − 0.999·20-s + 1.93·22-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.258 − 0.448i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0431 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0431 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.127763475\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.127763475\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 0.517iT - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.93T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.020651503315034008172141354013, −8.204307285978547925781263715826, −7.84153303911362572051173017641, −6.74560857693519716203142798479, −6.21107957952768793325261323976, −5.55167865777366586831507883055, −4.46993009587217488557714286044, −3.66380041736083309172590751153, −3.02552869422117859515680005404, −1.81947008087954444036104360104,
1.27205088101040315961842005909, 1.85659532200754494863825006509, 3.58450074654919258970586793849, 4.09572382895983578918584765421, 4.72951787677533256065230129799, 5.47616735864394888469979234232, 6.65131244222007698433363103554, 7.18278394097654887841978287623, 8.068865612520966156723464243804, 9.105115200751178729275752335472