L(s) = 1 | − 3-s + 0.732·5-s + 7-s + 9-s + 3.46·11-s − 4·13-s − 0.732·15-s − 6.19·17-s + 19-s − 21-s + 0.535·23-s − 4.46·25-s − 27-s + 4.19·29-s + 8.92·31-s − 3.46·33-s + 0.732·35-s − 0.535·37-s + 4·39-s − 2.53·41-s − 3.46·43-s + 0.732·45-s + 0.732·47-s + 49-s + 6.19·51-s + 5.26·53-s + 2.53·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.327·5-s + 0.377·7-s + 0.333·9-s + 1.04·11-s − 1.10·13-s − 0.189·15-s − 1.50·17-s + 0.229·19-s − 0.218·21-s + 0.111·23-s − 0.892·25-s − 0.192·27-s + 0.779·29-s + 1.60·31-s − 0.603·33-s + 0.123·35-s − 0.0881·37-s + 0.640·39-s − 0.396·41-s − 0.528·43-s + 0.109·45-s + 0.106·47-s + 0.142·49-s + 0.867·51-s + 0.723·53-s + 0.341·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.648822796\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.648822796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 23 | \( 1 - 0.535T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 0.732T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 2.92T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 - 0.196T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.072063092988033439727131666064, −7.07542486268669413064313404294, −6.64950284255965768803244196281, −6.00409022391890526108471468438, −5.03969043023911045835909285083, −4.58353450936060738986366608731, −3.79730279548064051110741373380, −2.57020932674238638439612322523, −1.83367136130467876464854791060, −0.68819536019624429886127449920,
0.68819536019624429886127449920, 1.83367136130467876464854791060, 2.57020932674238638439612322523, 3.79730279548064051110741373380, 4.58353450936060738986366608731, 5.03969043023911045835909285083, 6.00409022391890526108471468438, 6.64950284255965768803244196281, 7.07542486268669413064313404294, 8.072063092988033439727131666064