| L(s) = 1 | − 5.88·2-s − 265.·3-s − 477.·4-s − 2.55e3·5-s + 1.55e3·6-s − 3.87e3·7-s + 5.81e3·8-s + 5.06e4·9-s + 1.50e4·10-s − 2.34e4·11-s + 1.26e5·12-s + 2.43e4·13-s + 2.28e4·14-s + 6.77e5·15-s + 2.10e5·16-s − 2.97e5·18-s − 3.04e5·19-s + 1.21e6·20-s + 1.02e6·21-s + 1.38e5·22-s + 3.93e4·23-s − 1.54e6·24-s + 4.57e6·25-s − 1.43e5·26-s − 8.20e6·27-s + 1.85e6·28-s − 3.15e6·29-s + ⋯ |
| L(s) = 1 | − 0.259·2-s − 1.88·3-s − 0.932·4-s − 1.82·5-s + 0.491·6-s − 0.610·7-s + 0.502·8-s + 2.57·9-s + 0.475·10-s − 0.483·11-s + 1.76·12-s + 0.236·13-s + 0.158·14-s + 3.45·15-s + 0.801·16-s − 0.668·18-s − 0.536·19-s + 1.70·20-s + 1.15·21-s + 0.125·22-s + 0.0293·23-s − 0.949·24-s + 2.34·25-s − 0.0614·26-s − 2.97·27-s + 0.569·28-s − 0.828·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0002910376931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0002910376931\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 5.88T + 512T^{2} \) |
| 3 | \( 1 + 265.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.34e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.43e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 3.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.93e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.15e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.64e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.07e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 6.50e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.38e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.32e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.63e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.20e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.27e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.60e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.22e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.08e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.67e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.15e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.25e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.18e8T + 7.60e17T^{2} \) |
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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
−10.55540512673819695097337773467, −9.439311000829238869731711501087, −8.142482804681126334979242436130, −7.34059562347074495669410566871, −6.31323549019541844971792506210, −5.12911743925012083498295960962, −4.38674264194489455056731096855, −3.57859525552301881568364946942, −1.08411358661204034321888706467, −0.01108166210573308451408646797,
0.01108166210573308451408646797, 1.08411358661204034321888706467, 3.57859525552301881568364946942, 4.38674264194489455056731096855, 5.12911743925012083498295960962, 6.31323549019541844971792506210, 7.34059562347074495669410566871, 8.142482804681126334979242436130, 9.439311000829238869731711501087, 10.55540512673819695097337773467
