| L(s) = 1 | + 19.6·3-s − 25·5-s − 48.6·7-s + 142.·9-s − 283.·11-s + 169·13-s − 491.·15-s + 789.·17-s − 83.3·19-s − 955.·21-s + 1.93e3·23-s + 625·25-s − 1.96e3·27-s + 222.·29-s + 2.78e3·31-s − 5.56e3·33-s + 1.21e3·35-s + 8.36e3·37-s + 3.31e3·39-s + 1.21e3·41-s − 5.63e3·43-s − 3.57e3·45-s − 1.77e4·47-s − 1.44e4·49-s + 1.55e4·51-s + 1.08e4·53-s + 7.08e3·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 0.447·5-s − 0.375·7-s + 0.587·9-s − 0.706·11-s + 0.277·13-s − 0.563·15-s + 0.662·17-s − 0.0529·19-s − 0.472·21-s + 0.762·23-s + 0.200·25-s − 0.519·27-s + 0.0491·29-s + 0.521·31-s − 0.890·33-s + 0.167·35-s + 1.00·37-s + 0.349·39-s + 0.113·41-s − 0.465·43-s − 0.262·45-s − 1.17·47-s − 0.859·49-s + 0.834·51-s + 0.529·53-s + 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 3 | \( 1 - 19.6T + 243T^{2} \) |
| 7 | \( 1 + 48.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 283.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 789.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 83.3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 222.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.36e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.06e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.26e5T + 8.58e9T^{2} \) |
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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
−8.667113304752254855741038512910, −8.031015963332111296650441102436, −7.40707137502712305678876286074, −6.35750892470555127557458349317, −5.23887600500557711393328713686, −4.15075731845071741741851325060, −3.18633688900167520149547058053, −2.67721104588665522663585575467, −1.35510861283643087878898918155, 0,
1.35510861283643087878898918155, 2.67721104588665522663585575467, 3.18633688900167520149547058053, 4.15075731845071741741851325060, 5.23887600500557711393328713686, 6.35750892470555127557458349317, 7.40707137502712305678876286074, 8.031015963332111296650441102436, 8.667113304752254855741038512910
