| L(s) = 1 | − 3-s − 2i·5-s + 9-s − 6i·11-s + (−3 + 2i)13-s + 2i·15-s + 2·17-s − 8·23-s + 25-s − 27-s + 2·29-s + 8i·31-s + 6i·33-s + 8i·37-s + (3 − 2i)39-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894i·5-s + 0.333·9-s − 1.80i·11-s + (−0.832 + 0.554i)13-s + 0.516i·15-s + 0.485·17-s − 1.66·23-s + 0.200·25-s − 0.192·27-s + 0.371·29-s + 1.43i·31-s + 1.04i·33-s + 1.31i·37-s + (0.480 − 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9455131136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9455131136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.128228931845223254150444316238, −7.20361953267695860356047806305, −6.36821001260645792817847175920, −5.87097029531680364692354352131, −5.09286925517731839964961193611, −4.60193089821727371842453667894, −3.65251807467176059231258387984, −2.85978056359303672172976188363, −1.59748415582373900230186732222, −0.78416109741502704157906135794,
0.32063888126332614630463544135, 1.92530112759401155931296304154, 2.39548467631391986561937319550, 3.53330420591941084287388218704, 4.30312161031910766805076538676, 5.01144184304858452898634347383, 5.75431767410657427819405155796, 6.47135548916982895185261056872, 7.22257580550371470402193211286, 7.51254740816178573272467634155
