L(s) = 1 | + i·3-s + (2 + i)5-s − 2i·7-s − 9-s − 2·11-s + i·13-s + (−1 + 2i)15-s + 2i·17-s − 4·19-s + 2·21-s + (3 + 4i)25-s − i·27-s − 4·29-s − 8·31-s − 2i·33-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.894 + 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s + 0.277i·13-s + (−0.258 + 0.516i)15-s + 0.485i·17-s − 0.917·19-s + 0.436·21-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 0.742·29-s − 1.43·31-s − 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.020274996\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020274996\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 - i)T \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 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
−9.133268811439705425744653861659, −8.344736712617183097635433907133, −7.45040171591637541988342342090, −6.71509816029360762341239124059, −5.94947025571532300547259944484, −5.21686885616493223095949956927, −4.31890707998252421772462088689, −3.51500931558475101007930422432, −2.53329523386891776614274580468, −1.52987831900190524830934529347,
0.28491494573742795891265509846, 1.83546881909555737139101018036, 2.32225797222956469181091582995, 3.43088109584937124432054037674, 4.72234377214669647550459857644, 5.55896621300738060861822604438, 5.86624795008074368619694933757, 6.89687004795153531334743827776, 7.57750170309230984537305192346, 8.603798841262593642195515003299