| L(s) = 1 | − 0.806·3-s + 5-s − 1.19·7-s − 2.35·9-s − 4.15·11-s + 2.96·13-s − 0.806·15-s + 5.50·17-s + 3.19·19-s + 0.962·21-s − 1.84·23-s + 25-s + 4.31·27-s − 29-s + 4.80·31-s + 3.35·33-s − 1.19·35-s − 9.50·37-s − 2.38·39-s − 11.2·41-s + 0.0303·43-s − 2.35·45-s − 4.80·47-s − 5.57·49-s − 4.43·51-s − 1.35·53-s − 4.15·55-s + ⋯ |
| L(s) = 1 | − 0.465·3-s + 0.447·5-s − 0.451·7-s − 0.783·9-s − 1.25·11-s + 0.821·13-s − 0.208·15-s + 1.33·17-s + 0.732·19-s + 0.210·21-s − 0.384·23-s + 0.200·25-s + 0.829·27-s − 0.185·29-s + 0.863·31-s + 0.583·33-s − 0.201·35-s − 1.56·37-s − 0.382·39-s − 1.76·41-s + 0.00462·43-s − 0.350·45-s − 0.701·47-s − 0.796·49-s − 0.621·51-s − 0.185·53-s − 0.560·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 - 5.50T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 1.84T + 23T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 0.0303T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 + 5.84T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 4.93T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 + 1.38T + 97T^{2} \) |
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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
−8.479608262859881067241113218834, −7.990946730738580309494609549983, −6.95933501230260744099342894564, −6.10145999244611337077111590056, −5.50618569288743473641584916961, −4.93110585012273394135103594309, −3.42120148486559331899263347288, −2.89992172422406699571306929457, −1.47880296336066571406827809111, 0,
1.47880296336066571406827809111, 2.89992172422406699571306929457, 3.42120148486559331899263347288, 4.93110585012273394135103594309, 5.50618569288743473641584916961, 6.10145999244611337077111590056, 6.95933501230260744099342894564, 7.990946730738580309494609549983, 8.479608262859881067241113218834
