| L(s) = 1 | − 0.304·3-s − 35.5·5-s + 49·7-s − 242.·9-s − 565.·11-s − 983.·13-s + 10.8·15-s + 200.·17-s − 828.·19-s − 14.9·21-s + 4.43e3·23-s − 1.86e3·25-s + 147.·27-s + 3.71e3·29-s + 992.·31-s + 172.·33-s − 1.74e3·35-s + 8.35e3·37-s + 299.·39-s − 1.34e4·41-s − 298.·43-s + 8.62e3·45-s − 1.87e4·47-s + 2.40e3·49-s − 60.8·51-s − 1.60e4·53-s + 2.00e4·55-s + ⋯ |
| L(s) = 1 | − 0.0195·3-s − 0.635·5-s + 0.377·7-s − 0.999·9-s − 1.40·11-s − 1.61·13-s + 0.0123·15-s + 0.167·17-s − 0.526·19-s − 0.00737·21-s + 1.74·23-s − 0.596·25-s + 0.0390·27-s + 0.820·29-s + 0.185·31-s + 0.0275·33-s − 0.240·35-s + 1.00·37-s + 0.0314·39-s − 1.25·41-s − 0.0246·43-s + 0.635·45-s − 1.23·47-s + 0.142·49-s − 0.00327·51-s − 0.784·53-s + 0.895·55-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
| \(L(3)\) |
\(\approx\) |
\(0.8339346178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8339346178\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 0.304T + 243T^{2} \) |
| 5 | \( 1 + 35.5T + 3.12e3T^{2} \) |
| 11 | \( 1 + 565.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 983.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 200.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 828.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 992.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 298.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.87e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.11e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.26e4T + 8.58e9T^{2} \) |
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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
−10.40571021828409894317015255310, −9.396891433617168986716749183482, −8.213758067928144925089564866028, −7.78293382819531233055309323063, −6.67539905663485650503610405466, −5.26087859308627184330355461872, −4.76839062230300868954366574835, −3.16106685470675583174820220298, −2.33424930869627166099782869806, −0.44333013915077928038853183917,
0.44333013915077928038853183917, 2.33424930869627166099782869806, 3.16106685470675583174820220298, 4.76839062230300868954366574835, 5.26087859308627184330355461872, 6.67539905663485650503610405466, 7.78293382819531233055309323063, 8.213758067928144925089564866028, 9.396891433617168986716749183482, 10.40571021828409894317015255310
