| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s + 2·14-s − 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 2·21-s − 4·22-s + 8·23-s + 24-s + 25-s − 27-s − 2·28-s + 8·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.436·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 1.48·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.13298321787904, −13.72250888492061, −12.95896469330939, −12.59801807405246, −12.24300195459056, −11.59065716188148, −11.07774620995511, −10.63287429152999, −10.26387087131940, −9.513630549338089, −9.249867761101513, −8.819583697745550, −8.239834843294736, −7.452728149644089, −6.875029493440983, −6.496548398107963, −6.246130223891270, −5.503217796656991, −4.865705958134870, −4.199622899508867, −3.625739099816286, −2.777412127330119, −2.352328014532365, −1.299451227871678, −0.9740370203775657, 0,
0.9740370203775657, 1.299451227871678, 2.352328014532365, 2.777412127330119, 3.625739099816286, 4.199622899508867, 4.865705958134870, 5.503217796656991, 6.246130223891270, 6.496548398107963, 6.875029493440983, 7.452728149644089, 8.239834843294736, 8.819583697745550, 9.249867761101513, 9.513630549338089, 10.26387087131940, 10.63287429152999, 11.07774620995511, 11.59065716188148, 12.24300195459056, 12.59801807405246, 12.95896469330939, 13.72250888492061, 14.13298321787904
