L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·9-s − 2·11-s + 14-s + 16-s + 17-s − 3·18-s − 2·19-s − 2·22-s + 8·23-s + 28-s + 8·31-s + 32-s + 34-s − 3·36-s + 4·37-s − 2·38-s − 6·41-s − 4·43-s − 2·44-s + 8·46-s − 8·47-s + 49-s + 6·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 9-s − 0.603·11-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 0.458·19-s − 0.426·22-s + 1.66·23-s + 0.188·28-s + 1.43·31-s + 0.176·32-s + 0.171·34-s − 1/2·36-s + 0.657·37-s − 0.324·38-s − 0.937·41-s − 0.609·43-s − 0.301·44-s + 1.17·46-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.988052889\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.988052889\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.287246260120344010314079645842, −7.23931326327263639717983865302, −6.59958049753719151252988153357, −5.83333020221613009374084113125, −5.10172949323264174123490289308, −4.68536371262132186963410815769, −3.54929293043017424167232544617, −2.88326533815166094431135424089, −2.13240357511273868760525630474, −0.809988454735605141576757027849,
0.809988454735605141576757027849, 2.13240357511273868760525630474, 2.88326533815166094431135424089, 3.54929293043017424167232544617, 4.68536371262132186963410815769, 5.10172949323264174123490289308, 5.83333020221613009374084113125, 6.59958049753719151252988153357, 7.23931326327263639717983865302, 8.287246260120344010314079645842