| L(s) = 1 | − 5-s − 13-s + 4·17-s + 6·19-s − 8·23-s + 25-s − 6·29-s − 6·31-s − 12·37-s − 6·41-s + 2·43-s + 6·47-s − 6·53-s + 8·61-s + 65-s − 4·67-s − 12·71-s + 2·73-s + 6·83-s − 4·85-s − 18·89-s − 6·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.277·13-s + 0.970·17-s + 1.37·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.07·31-s − 1.97·37-s − 0.937·41-s + 0.304·43-s + 0.875·47-s − 0.824·53-s + 1.02·61-s + 0.124·65-s − 0.488·67-s − 1.42·71-s + 0.234·73-s + 0.658·83-s − 0.433·85-s − 1.90·89-s − 0.615·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6843769933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6843769933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.86777492468632, −12.22685852506124, −12.12581822736501, −11.64516930823211, −11.14045519692775, −10.56185963647781, −10.10873562966783, −9.733135518925649, −9.210740780410507, −8.704443083190911, −8.153724603859895, −7.664908330246784, −7.285572231827633, −6.965874119662980, −6.085203569493425, −5.681612646285280, −5.251029565396904, −4.764508794342764, −3.900496333109093, −3.640412770908772, −3.185544923373281, −2.395847092356258, −1.728946743274659, −1.238793872743539, −0.2304644475200840,
0.2304644475200840, 1.238793872743539, 1.728946743274659, 2.395847092356258, 3.185544923373281, 3.640412770908772, 3.900496333109093, 4.764508794342764, 5.251029565396904, 5.681612646285280, 6.085203569493425, 6.965874119662980, 7.285572231827633, 7.664908330246784, 8.153724603859895, 8.704443083190911, 9.210740780410507, 9.733135518925649, 10.10873562966783, 10.56185963647781, 11.14045519692775, 11.64516930823211, 12.12581822736501, 12.22685852506124, 12.86777492468632
