L(s) = 1 | − 169. i·2-s − 817. i·3-s + 3.90e3·4-s + 5.86e4·5-s − 1.38e5·6-s − 2.72e6·7-s − 6.23e6i·8-s + 1.36e7·9-s − 9.96e6i·10-s + 4.12e7i·11-s − 3.19e6i·12-s − 1.27e8·13-s + 4.63e8i·14-s − 4.79e7i·15-s − 9.30e8·16-s − 1.09e9i·17-s + ⋯ |
L(s) = 1 | − 0.938i·2-s − 0.215i·3-s + 0.119·4-s + 0.335·5-s − 0.202·6-s − 1.25·7-s − 1.05i·8-s + 0.953·9-s − 0.315i·10-s + 0.637i·11-s − 0.0257i·12-s − 0.562·13-s + 1.17i·14-s − 0.0725i·15-s − 0.866·16-s − 0.650i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(0.4083207883\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4083207883\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (3.62e10 + 8.55e10i)T \) |
good | 2 | \( 1 + 169. iT - 3.27e4T^{2} \) |
| 3 | \( 1 + 817. iT - 1.43e7T^{2} \) |
| 5 | \( 1 - 5.86e4T + 3.05e10T^{2} \) |
| 7 | \( 1 + 2.72e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 4.12e7iT - 4.17e15T^{2} \) |
| 13 | \( 1 + 1.27e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.09e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 + 1.07e9iT - 1.51e19T^{2} \) |
| 23 | \( 1 + 2.61e10T + 2.66e20T^{2} \) |
| 31 | \( 1 - 8.60e10iT - 2.34e22T^{2} \) |
| 37 | \( 1 - 8.70e11iT - 3.33e23T^{2} \) |
| 41 | \( 1 + 2.48e11iT - 1.55e24T^{2} \) |
| 43 | \( 1 - 1.01e12iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 1.95e12iT - 1.20e25T^{2} \) |
| 53 | \( 1 + 5.78e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.84e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.21e13iT - 6.02e26T^{2} \) |
| 67 | \( 1 + 5.47e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 3.27e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 9.03e13iT - 8.90e27T^{2} \) |
| 79 | \( 1 - 3.62e12iT - 2.91e28T^{2} \) |
| 83 | \( 1 - 9.36e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 3.62e14iT - 1.74e29T^{2} \) |
| 97 | \( 1 + 1.39e15iT - 6.33e29T^{2} \) |
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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
−12.66350215178741350344848925610, −11.78970360461099710901593835396, −9.954342300323250929457286084371, −9.794387886096071798422916419930, −7.35803966199431082908175421307, −6.30545252906668045151607765978, −4.19110728364692960543487297707, −2.76074235092084647046020230613, −1.63852429401715641256515263314, −0.10236805342126160635464023765,
2.03854177820724288674112869293, 3.76485332550176129606422067341, 5.64424936450703330109190629841, 6.57628596413418257649922269937, 7.79873442720040739558805673592, 9.420884404574274509428122364491, 10.52407524463418890637522144622, 12.25013938391420285712509385719, 13.49307975950637607525037593912, 14.77263167262153652860436170556