| L(s) = 1 | + 0.732·3-s − 0.732·7-s − 2.46·9-s − 4.73·11-s + 1.46·13-s + 17-s + 5.46·19-s − 0.535·21-s + 4.73·23-s − 4·27-s − 3.46·29-s − 6.19·31-s − 3.46·33-s − 11.4·37-s + 1.07·39-s − 6·41-s − 12.3·43-s − 6.92·47-s − 6.46·49-s + 0.732·51-s + 0.928·53-s + 4·57-s + 9.46·59-s − 7.46·61-s + 1.80·63-s − 1.07·67-s + 3.46·69-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 0.276·7-s − 0.821·9-s − 1.42·11-s + 0.406·13-s + 0.242·17-s + 1.25·19-s − 0.116·21-s + 0.986·23-s − 0.769·27-s − 0.643·29-s − 1.11·31-s − 0.603·33-s − 1.88·37-s + 0.171·39-s − 0.937·41-s − 1.88·43-s − 1.01·47-s − 0.923·49-s + 0.102·51-s + 0.127·53-s + 0.529·57-s + 1.23·59-s − 0.955·61-s + 0.227·63-s − 0.130·67-s + 0.417·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 6.19T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 + 7.46T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 1.80T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−8.864556791434598050040105045626, −8.203349552705990508041891198792, −7.47502844956944497549258828321, −6.62819792945162147646545099871, −5.34355101471231681499046908953, −5.20603820853657259494578215717, −3.43418740288784399731943611540, −3.10336844100511539339169604309, −1.80479644723143014792244333759, 0,
1.80479644723143014792244333759, 3.10336844100511539339169604309, 3.43418740288784399731943611540, 5.20603820853657259494578215717, 5.34355101471231681499046908953, 6.62819792945162147646545099871, 7.47502844956944497549258828321, 8.203349552705990508041891198792, 8.864556791434598050040105045626
