| L(s) = 1 | + 3·3-s + 5-s + 2·7-s + 6·9-s − 11-s + 3·13-s + 3·15-s + 17-s + 7·19-s + 6·21-s + 2·23-s + 25-s + 9·27-s − 29-s + 7·31-s − 3·33-s + 2·35-s − 8·37-s + 9·39-s − 2·41-s + 6·43-s + 6·45-s − 7·47-s − 3·49-s + 3·51-s − 11·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.755·7-s + 2·9-s − 0.301·11-s + 0.832·13-s + 0.774·15-s + 0.242·17-s + 1.60·19-s + 1.30·21-s + 0.417·23-s + 1/5·25-s + 1.73·27-s − 0.185·29-s + 1.25·31-s − 0.522·33-s + 0.338·35-s − 1.31·37-s + 1.44·39-s − 0.312·41-s + 0.914·43-s + 0.894·45-s − 1.02·47-s − 3/7·49-s + 0.420·51-s − 1.51·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.003208401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.003208401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−14.22076127967177, −13.85253945897136, −13.59598200508896, −12.96884275610947, −12.48642654998824, −11.80183827800602, −11.20689352418780, −10.68174312386076, −9.998767534102831, −9.614207257001250, −9.151248286083191, −8.561940956937969, −8.125002597112873, −7.772952979948884, −7.173070431499548, −6.579208144196098, −5.824759852704733, −5.079524439790911, −4.684887921397516, −3.863154592981284, −3.133082991338492, −3.036536506651671, −2.026596322160788, −1.574897002228644, −0.9171885672407217,
0.9171885672407217, 1.574897002228644, 2.026596322160788, 3.036536506651671, 3.133082991338492, 3.863154592981284, 4.684887921397516, 5.079524439790911, 5.824759852704733, 6.579208144196098, 7.173070431499548, 7.772952979948884, 8.125002597112873, 8.561940956937969, 9.151248286083191, 9.614207257001250, 9.998767534102831, 10.68174312386076, 11.20689352418780, 11.80183827800602, 12.48642654998824, 12.96884275610947, 13.59598200508896, 13.85253945897136, 14.22076127967177
