| L(s) = 1 | − 32.5·2-s − 213.·3-s + 547.·4-s + 2.40e3·5-s + 6.94e3·6-s + 8.34e3·7-s − 1.14e3·8-s + 2.58e4·9-s − 7.83e4·10-s − 2.94e4·11-s − 1.16e5·12-s + 1.14e5·13-s − 2.71e5·14-s − 5.13e5·15-s − 2.42e5·16-s − 8.40e5·18-s + 7.91e4·19-s + 1.31e6·20-s − 1.78e6·21-s + 9.57e5·22-s + 2.58e6·23-s + 2.43e5·24-s + 3.83e6·25-s − 3.73e6·26-s − 1.31e6·27-s + 4.56e6·28-s + 5.69e5·29-s + ⋯ |
| L(s) = 1 | − 1.43·2-s − 1.52·3-s + 1.06·4-s + 1.72·5-s + 2.18·6-s + 1.31·7-s − 0.0986·8-s + 1.31·9-s − 2.47·10-s − 0.605·11-s − 1.62·12-s + 1.11·13-s − 1.89·14-s − 2.61·15-s − 0.926·16-s − 1.88·18-s + 0.139·19-s + 1.84·20-s − 1.99·21-s + 0.870·22-s + 1.92·23-s + 0.150·24-s + 1.96·25-s − 1.60·26-s − 0.475·27-s + 1.40·28-s + 0.149·29-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
| \(L(5)\) |
\(\approx\) |
\(1.326583688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.326583688\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 32.5T + 512T^{2} \) |
| 3 | \( 1 + 213.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.40e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.34e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.94e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.14e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.91e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.58e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.69e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.28e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.35e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.82e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.20e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.37e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.36e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.58e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.38e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.34e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.85e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 9.57e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.11e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.05e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.08e8T + 7.60e17T^{2} \) |
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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
−10.39400001368284580554172129162, −9.373521055293012143499686510188, −8.565287117377419016712680270042, −7.35366911619087766487120744843, −6.34311878086639010797666401672, −5.47144481751770327743851976741, −4.78279759438921814192076978784, −2.30484136642505250364266150015, −1.26165389292012219348114093638, −0.851096519639918461437654493234,
0.851096519639918461437654493234, 1.26165389292012219348114093638, 2.30484136642505250364266150015, 4.78279759438921814192076978784, 5.47144481751770327743851976741, 6.34311878086639010797666401672, 7.35366911619087766487120744843, 8.565287117377419016712680270042, 9.373521055293012143499686510188, 10.39400001368284580554172129162
