| L(s) = 1 | + 2·3-s + 96·5-s + 49·7-s − 239·9-s + 720·11-s − 572·13-s + 192·15-s + 1.25e3·17-s + 94·19-s + 98·21-s + 96·23-s + 6.09e3·25-s − 964·27-s + 4.37e3·29-s − 6.24e3·31-s + 1.44e3·33-s + 4.70e3·35-s + 1.07e4·37-s − 1.14e3·39-s + 1.20e4·41-s + 9.16e3·43-s − 2.29e4·45-s − 2.58e4·47-s + 2.40e3·49-s + 2.50e3·51-s − 1.01e3·53-s + 6.91e4·55-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 1.71·5-s + 0.377·7-s − 0.983·9-s + 1.79·11-s − 0.938·13-s + 0.220·15-s + 1.05·17-s + 0.0597·19-s + 0.0484·21-s + 0.0378·23-s + 1.94·25-s − 0.254·27-s + 0.965·29-s − 1.16·31-s + 0.230·33-s + 0.649·35-s + 1.29·37-s − 0.120·39-s + 1.11·41-s + 0.755·43-s − 1.68·45-s − 1.70·47-s + 1/7·49-s + 0.135·51-s − 0.0495·53-s + 3.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.604344163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.604344163\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 2 T + p^{5} T^{2} \) |
| 5 | \( 1 - 96 T + p^{5} T^{2} \) |
| 11 | \( 1 - 720 T + p^{5} T^{2} \) |
| 13 | \( 1 + 44 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1254 T + p^{5} T^{2} \) |
| 19 | \( 1 - 94 T + p^{5} T^{2} \) |
| 23 | \( 1 - 96 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4374 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6244 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10798 T + p^{5} T^{2} \) |
| 41 | \( 1 - 12006 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9160 T + p^{5} T^{2} \) |
| 47 | \( 1 + 25836 T + p^{5} T^{2} \) |
| 53 | \( 1 + 1014 T + p^{5} T^{2} \) |
| 59 | \( 1 + 1242 T + p^{5} T^{2} \) |
| 61 | \( 1 + 7592 T + p^{5} T^{2} \) |
| 67 | \( 1 + 41132 T + p^{5} T^{2} \) |
| 71 | \( 1 + 37632 T + p^{5} T^{2} \) |
| 73 | \( 1 + 13438 T + p^{5} T^{2} \) |
| 79 | \( 1 - 6248 T + p^{5} T^{2} \) |
| 83 | \( 1 - 25254 T + p^{5} T^{2} \) |
| 89 | \( 1 + 45126 T + p^{5} T^{2} \) |
| 97 | \( 1 - 107222 T + p^{5} T^{2} \) |
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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
−10.05069876965832562943581152896, −9.403014582641006545169654619137, −8.792543795973682342314229813791, −7.49491500465372427353718688073, −6.29393215778616401877033476394, −5.75190589319767868998713224294, −4.66260390706071764405195568343, −3.13011828055702580967506274545, −2.04591291621398054278624469522, −1.04664351522192062629010364183,
1.04664351522192062629010364183, 2.04591291621398054278624469522, 3.13011828055702580967506274545, 4.66260390706071764405195568343, 5.75190589319767868998713224294, 6.29393215778616401877033476394, 7.49491500465372427353718688073, 8.792543795973682342314229813791, 9.403014582641006545169654619137, 10.05069876965832562943581152896
