L(s) = 1 | − 2·4-s − 5-s − 7-s + 4·13-s + 4·16-s + 17-s − 6·19-s + 2·20-s − 6·23-s − 4·25-s + 2·28-s + 29-s + 4·31-s + 35-s − 6·37-s + 6·41-s + 5·43-s − 7·47-s + 49-s − 8·52-s + 4·53-s + 11·59-s + 8·61-s − 8·64-s − 4·65-s + 67-s − 2·68-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s + 1.10·13-s + 16-s + 0.242·17-s − 1.37·19-s + 0.447·20-s − 1.25·23-s − 4/5·25-s + 0.377·28-s + 0.185·29-s + 0.718·31-s + 0.169·35-s − 0.986·37-s + 0.937·41-s + 0.762·43-s − 1.02·47-s + 1/7·49-s − 1.10·52-s + 0.549·53-s + 1.43·59-s + 1.02·61-s − 64-s − 0.496·65-s + 0.122·67-s − 0.242·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9471182972\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9471182972\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.07495204345577, −12.42153168408561, −12.20105328457794, −11.67845340900005, −10.93165673942756, −10.72790207799277, −10.06833061010079, −9.668278457813231, −9.299311562798085, −8.453606505474923, −8.356934355833629, −8.066720391137851, −7.296356000108568, −6.633771209916208, −6.284132632897838, −5.596757988718451, −5.385058184037105, −4.429730598716783, −4.048450268518349, −3.859706938179202, −3.191930037823334, −2.434211428885868, −1.773111099695980, −0.9756434596537157, −0.3273156854736265,
0.3273156854736265, 0.9756434596537157, 1.773111099695980, 2.434211428885868, 3.191930037823334, 3.859706938179202, 4.048450268518349, 4.429730598716783, 5.385058184037105, 5.596757988718451, 6.284132632897838, 6.633771209916208, 7.296356000108568, 8.066720391137851, 8.356934355833629, 8.453606505474923, 9.299311562798085, 9.668278457813231, 10.06833061010079, 10.72790207799277, 10.93165673942756, 11.67845340900005, 12.20105328457794, 12.42153168408561, 13.07495204345577