| L(s) = 1 | − 3·5-s − 7-s + 13-s − 5·17-s − 4·19-s + 4·23-s + 4·25-s + 7·29-s − 4·31-s + 3·35-s − 5·37-s − 9·41-s + 49-s − 3·53-s + 2·61-s − 3·65-s + 8·67-s + 6·73-s + 15·85-s + 13·89-s − 91-s + 12·95-s − 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 0.277·13-s − 1.21·17-s − 0.917·19-s + 0.834·23-s + 4/5·25-s + 1.29·29-s − 0.718·31-s + 0.507·35-s − 0.821·37-s − 1.40·41-s + 1/7·49-s − 0.412·53-s + 0.256·61-s − 0.372·65-s + 0.977·67-s + 0.702·73-s + 1.62·85-s + 1.37·89-s − 0.104·91-s + 1.23·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.79057660879984, −13.96354201244825, −13.52733992216730, −12.92825207243562, −12.55382403993092, −11.96708963875910, −11.56928432151139, −10.90005621633281, −10.72002826410773, −10.03316132802229, −9.278725005235848, −8.776766206705312, −8.357151283595622, −7.936523521967934, −7.104250533639562, −6.772874315882011, −6.371856740719255, −5.458435222146220, −4.790311776035320, −4.367715971397635, −3.685376464135956, −3.286922401278732, −2.490251121338398, −1.743234302647602, −0.6931472250502738, 0,
0.6931472250502738, 1.743234302647602, 2.490251121338398, 3.286922401278732, 3.685376464135956, 4.367715971397635, 4.790311776035320, 5.458435222146220, 6.371856740719255, 6.772874315882011, 7.104250533639562, 7.936523521967934, 8.357151283595622, 8.776766206705312, 9.278725005235848, 10.03316132802229, 10.72002826410773, 10.90005621633281, 11.56928432151139, 11.96708963875910, 12.55382403993092, 12.92825207243562, 13.52733992216730, 13.96354201244825, 14.79057660879984
