| L(s) = 1 | + 16.1·2-s + 264.·3-s − 252.·4-s + 601.·5-s + 4.25e3·6-s + 1.06e3·7-s − 1.23e4·8-s + 5.02e4·9-s + 9.69e3·10-s − 1.78e4·11-s − 6.68e4·12-s + 3.77e4·13-s + 1.71e4·14-s + 1.59e5·15-s − 6.88e4·16-s + 8.09e5·18-s − 1.72e5·19-s − 1.52e5·20-s + 2.80e5·21-s − 2.87e5·22-s + 2.52e6·23-s − 3.25e6·24-s − 1.59e6·25-s + 6.07e5·26-s + 8.09e6·27-s − 2.68e5·28-s − 3.17e6·29-s + ⋯ |
| L(s) = 1 | + 0.711·2-s + 1.88·3-s − 0.493·4-s + 0.430·5-s + 1.34·6-s + 0.167·7-s − 1.06·8-s + 2.55·9-s + 0.306·10-s − 0.367·11-s − 0.930·12-s + 0.366·13-s + 0.118·14-s + 0.812·15-s − 0.262·16-s + 1.81·18-s − 0.303·19-s − 0.212·20-s + 0.315·21-s − 0.261·22-s + 1.87·23-s − 2.00·24-s − 0.814·25-s + 0.260·26-s + 2.93·27-s − 0.0825·28-s − 0.833·29-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
| \(L(5)\) |
\(\approx\) |
\(6.967616781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.967616781\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 16.1T + 512T^{2} \) |
| 3 | \( 1 - 264.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 601.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.06e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.78e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.77e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.72e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.52e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.69e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.35e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.57e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.63e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.00e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.84e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.11e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.72e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.58e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.68e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.49e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.54e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.78e8T + 7.60e17T^{2} \) |
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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
−9.754294720818662391892582664981, −9.277436859580604051523506576129, −8.404745753495205273905107328914, −7.61934766835421448667455717796, −6.29088549512845709535553136037, −4.92545757502668552490969702902, −4.03983605927725248209043620046, −3.09757671729284542675274503020, −2.32909011406286811757673570392, −1.01371005157450744929699638326,
1.01371005157450744929699638326, 2.32909011406286811757673570392, 3.09757671729284542675274503020, 4.03983605927725248209043620046, 4.92545757502668552490969702902, 6.29088549512845709535553136037, 7.61934766835421448667455717796, 8.404745753495205273905107328914, 9.277436859580604051523506576129, 9.754294720818662391892582664981
