| L(s) = 1 | − 3-s + 3·7-s + 9-s + 2·11-s + 5·13-s + 17-s + 7·19-s − 3·21-s − 4·23-s − 27-s − 4·29-s − 5·31-s − 2·33-s − 6·37-s − 5·39-s + 2·41-s + 5·43-s + 12·47-s + 2·49-s − 51-s − 6·53-s − 7·57-s − 10·59-s + 13·61-s + 3·63-s + 5·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.603·11-s + 1.38·13-s + 0.242·17-s + 1.60·19-s − 0.654·21-s − 0.834·23-s − 0.192·27-s − 0.742·29-s − 0.898·31-s − 0.348·33-s − 0.986·37-s − 0.800·39-s + 0.312·41-s + 0.762·43-s + 1.75·47-s + 2/7·49-s − 0.140·51-s − 0.824·53-s − 0.927·57-s − 1.30·59-s + 1.66·61-s + 0.377·63-s + 0.610·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.720952876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.720952876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.83829357338400, −15.15221967469137, −14.40715555359242, −13.99052838166346, −13.67430160336645, −12.79340392598051, −12.23930514959581, −11.71835870182537, −11.17525222679920, −10.91886163834379, −10.20689991848970, −9.391083324224935, −9.037263673099344, −8.208764890496551, −7.740198531375486, −7.145936343792368, −6.426012129517545, −5.584289436269852, −5.487259704726333, −4.548749997664092, −3.846240364705613, −3.385265903035296, −2.101670571451908, −1.437970370769223, −0.7792807896602335,
0.7792807896602335, 1.437970370769223, 2.101670571451908, 3.385265903035296, 3.846240364705613, 4.548749997664092, 5.487259704726333, 5.584289436269852, 6.426012129517545, 7.145936343792368, 7.740198531375486, 8.208764890496551, 9.037263673099344, 9.391083324224935, 10.20689991848970, 10.91886163834379, 11.17525222679920, 11.71835870182537, 12.23930514959581, 12.79340392598051, 13.67430160336645, 13.99052838166346, 14.40715555359242, 15.15221967469137, 15.83829357338400
