| L(s) = 1 | + 2-s − 2·3-s + 4-s − 4·5-s − 2·6-s − 7-s + 8-s + 9-s − 4·10-s − 3·11-s − 2·12-s − 13-s − 14-s + 8·15-s + 16-s − 3·17-s + 18-s + 4·19-s − 4·20-s + 2·21-s − 3·22-s + 23-s − 2·24-s + 11·25-s − 26-s + 4·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.904·11-s − 0.577·12-s − 0.277·13-s − 0.267·14-s + 2.06·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.917·19-s − 0.894·20-s + 0.436·21-s − 0.639·22-s + 0.208·23-s − 0.408·24-s + 11/5·25-s − 0.196·26-s + 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 1171 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.76630215091218, −15.26672098867399, −14.74336409818095, −14.10138768181290, −13.24549980774905, −12.98577131984249, −12.20129519727695, −12.03279783564673, −11.45425121034355, −11.04314805665527, −10.64772496605439, −9.992227806630496, −9.110628493787345, −8.436875726774632, −7.769465987882686, −7.334900269248973, −6.809743177388793, −6.210725685704196, −5.418669379804101, −4.958790362780864, −4.555462255287266, −3.661378912298951, −3.278787208498678, −2.502374481485581, −1.272596512113892, 0, 0,
1.272596512113892, 2.502374481485581, 3.278787208498678, 3.661378912298951, 4.555462255287266, 4.958790362780864, 5.418669379804101, 6.210725685704196, 6.809743177388793, 7.334900269248973, 7.769465987882686, 8.436875726774632, 9.110628493787345, 9.992227806630496, 10.64772496605439, 11.04314805665527, 11.45425121034355, 12.03279783564673, 12.20129519727695, 12.98577131984249, 13.24549980774905, 14.10138768181290, 14.74336409818095, 15.26672098867399, 15.76630215091218
