L(s) = 1 | + (9.20 + 90.0i)2-s + (618. + 1.10e3i)3-s + (−8.02e3 + 1.65e3i)4-s − 4.92e4i·5-s + (−9.34e4 + 6.58e4i)6-s − 5.27e5i·7-s + (−2.23e5 − 7.07e5i)8-s + (−8.28e5 + 1.36e6i)9-s + (4.43e6 − 4.52e5i)10-s − 4.09e6·11-s + (−6.78e6 − 7.80e6i)12-s − 8.75e6·13-s + (4.75e7 − 4.85e6i)14-s + (5.41e7 − 3.04e7i)15-s + (6.16e7 − 2.65e7i)16-s + 5.78e7i·17-s + ⋯ |
L(s) = 1 | + (0.101 + 0.994i)2-s + (0.490 + 0.871i)3-s + (−0.979 + 0.202i)4-s − 1.40i·5-s + (−0.817 + 0.576i)6-s − 1.69i·7-s + (−0.300 − 0.953i)8-s + (−0.519 + 0.854i)9-s + (1.40 − 0.143i)10-s − 0.697·11-s + (−0.656 − 0.754i)12-s − 0.503·13-s + (1.68 − 0.172i)14-s + (1.22 − 0.690i)15-s + (0.918 − 0.396i)16-s + 0.581i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(1.01613 - 0.462921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01613 - 0.462921i\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-9.20 - 90.0i)T \) |
| 3 | \( 1 + (-618. - 1.10e3i)T \) |
good | 5 | \( 1 + 4.92e4iT - 1.22e9T^{2} \) |
| 7 | \( 1 + 5.27e5iT - 9.68e10T^{2} \) |
| 11 | \( 1 + 4.09e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 8.75e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 5.78e7iT - 9.90e15T^{2} \) |
| 19 | \( 1 + 2.75e8iT - 4.20e16T^{2} \) |
| 23 | \( 1 + 4.87e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.09e9iT - 1.02e19T^{2} \) |
| 31 | \( 1 + 7.35e8iT - 2.44e19T^{2} \) |
| 37 | \( 1 - 8.45e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.02e10iT - 9.25e20T^{2} \) |
| 43 | \( 1 + 6.24e10iT - 1.71e21T^{2} \) |
| 47 | \( 1 - 5.46e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 4.50e10iT - 2.60e22T^{2} \) |
| 59 | \( 1 - 1.17e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 1.79e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 2.45e11iT - 5.48e23T^{2} \) |
| 71 | \( 1 - 5.79e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.56e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.18e12iT - 4.66e24T^{2} \) |
| 83 | \( 1 + 1.53e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 2.56e12iT - 2.19e25T^{2} \) |
| 97 | \( 1 + 3.42e12T + 6.73e25T^{2} \) |
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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
−16.61909101471773241927663398317, −15.57539652974228514724600631448, −13.97327051280261121200186125478, −13.02251306866285464402859034003, −10.28823966272968875429424500590, −8.887792077159156297007292937752, −7.57943003617014474915948132728, −5.04948755415200233556706754858, −4.06421429349565995895384592223, −0.42440419268417439284258730690,
2.18191757605548171907609549257, 2.97325958978046321741490911405, 5.92889800489479981332693829869, 8.044826488268819162519357198645, 9.737572443894631020484994993081, 11.52556839454985921805001427244, 12.56215986321366279930896566627, 14.16580962019332582496383768491, 15.08323324005761330367120025084, 18.18494174118267885888943790687