| L(s) = 1 | + 45.2·2-s + 473. i·3-s + 2.04e3·4-s + 2.10e4i·5-s + 2.14e4i·6-s + 9.26e4·8-s + 3.07e5·9-s + 9.52e5i·10-s + 4.93e5·11-s + 9.68e5i·12-s + 3.20e6i·13-s − 9.95e6·15-s + 4.19e6·16-s + 3.44e7i·17-s + 1.39e7·18-s − 2.13e7i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.648i·3-s + 0.500·4-s + 1.34i·5-s + 0.458i·6-s + 0.353·8-s + 0.578·9-s + 0.952i·10-s + 0.278·11-s + 0.324i·12-s + 0.664i·13-s − 0.873·15-s + 0.250·16-s + 1.42i·17-s + 0.409·18-s − 0.453i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(3.856162567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.856162567\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 45.2T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 473. iT - 5.31e5T^{2} \) |
| 5 | \( 1 - 2.10e4iT - 2.44e8T^{2} \) |
| 11 | \( 1 - 4.93e5T + 3.13e12T^{2} \) |
| 13 | \( 1 - 3.20e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 - 3.44e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 2.13e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 - 2.45e8T + 2.19e16T^{2} \) |
| 29 | \( 1 - 8.71e8T + 3.53e17T^{2} \) |
| 31 | \( 1 - 7.11e8iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 2.45e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 4.29e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 8.71e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + 5.70e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 - 9.55e9T + 4.91e20T^{2} \) |
| 59 | \( 1 + 5.58e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 6.90e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 + 7.74e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + 1.07e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + 2.39e10iT - 2.29e22T^{2} \) |
| 79 | \( 1 + 3.19e10T + 5.90e22T^{2} \) |
| 83 | \( 1 + 3.39e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 - 8.37e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 9.93e11iT - 6.93e23T^{2} \) |
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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
−11.86486061863940397174770967779, −10.72102079972608558547867254474, −10.28866623762322976134388197926, −8.801394623545720219654565271830, −7.07106589339745242467871254723, −6.54194287421859914649105943461, −4.98623927961002617448264869856, −3.86568326715732114996621835913, −2.97892776260083650976797434485, −1.57596928434412636059536988177,
0.69263638610161031301003311005, 1.42003956923301557932045196663, 2.93143181423637719953178257881, 4.50172252087309580242059267928, 5.23654449682700181819630918365, 6.63202175133266702131672982114, 7.68172542704707982327021817866, 8.838540020874748628776169291562, 10.05621924098057650627202732335, 11.59821053562539113321920128458
