L(s) = 1 | + (−2 − 2i)3-s + (2 − i)5-s + (2 − 2i)7-s + 5i·9-s + (1 − i)13-s + (−6 − 2i)15-s + (−5 − 5i)17-s − 4·19-s − 8·21-s + (2 + 2i)23-s + (3 − 4i)25-s + (4 − 4i)27-s + 4i·29-s − 4i·31-s + (2 − 6i)35-s + ⋯ |
L(s) = 1 | + (−1.15 − 1.15i)3-s + (0.894 − 0.447i)5-s + (0.755 − 0.755i)7-s + 1.66i·9-s + (0.277 − 0.277i)13-s + (−1.54 − 0.516i)15-s + (−1.21 − 1.21i)17-s − 0.917·19-s − 1.74·21-s + (0.417 + 0.417i)23-s + (0.600 − 0.800i)25-s + (0.769 − 0.769i)27-s + 0.742i·29-s − 0.718i·31-s + (0.338 − 1.01i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.496521 - 0.890563i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.496521 - 0.890563i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
good | 3 | \( 1 + (2 + 2i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (5 + 5i)T + 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-2 - 2i)T + 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + (2 - 2i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7 + 7i)T - 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + (-10 + 10i)T - 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (3 - 3i)T - 73iT^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + (-2 - 2i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−11.17537145608867586345175392932, −10.86126265487268488627904125241, −9.497375091892541281233676851995, −8.310363181742190250595575054547, −7.20767670096183399373053095905, −6.47976866008577400329105873132, −5.44125418197532702960391007894, −4.58041243288640931632434359529, −2.13202886647570508909171568080, −0.873977880245997273451414525901,
2.14675616168741467365200642355, 4.05574374549856144630671210715, 5.04727876082139398117591533819, 5.94493156480446112546251983154, 6.63658903681055329547280022023, 8.531854203808900664650264952255, 9.261371095628691839644381630784, 10.49161607431376272760035512623, 10.76419019559640746208847008356, 11.65259582350454511875998243899