| L(s) = 1 | + 10·2-s − 73·3-s − 28·4-s − 295·5-s − 730·6-s + 1.37e3·7-s − 1.56e3·8-s + 3.14e3·9-s − 2.95e3·10-s − 7.64e3·11-s + 2.04e3·12-s + 2.19e3·13-s + 1.37e4·14-s + 2.15e4·15-s − 1.20e4·16-s − 4.14e3·17-s + 3.14e4·18-s − 3.18e3·19-s + 8.26e3·20-s − 1.00e5·21-s − 7.64e4·22-s − 1.77e4·23-s + 1.13e5·24-s + 8.90e3·25-s + 2.19e4·26-s − 6.97e4·27-s − 3.84e4·28-s + ⋯ |
| L(s) = 1 | + 0.883·2-s − 1.56·3-s − 0.218·4-s − 1.05·5-s − 1.37·6-s + 1.51·7-s − 1.07·8-s + 1.43·9-s − 0.932·10-s − 1.73·11-s + 0.341·12-s + 0.277·13-s + 1.33·14-s + 1.64·15-s − 0.733·16-s − 0.204·17-s + 1.26·18-s − 0.106·19-s + 0.230·20-s − 2.36·21-s − 1.53·22-s − 0.304·23-s + 1.68·24-s + 0.113·25-s + 0.245·26-s − 0.681·27-s − 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 - 5 p T + p^{7} T^{2} \) |
| 3 | \( 1 + 73 T + p^{7} T^{2} \) |
| 5 | \( 1 + 59 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 1373 T + p^{7} T^{2} \) |
| 11 | \( 1 + 7646 T + p^{7} T^{2} \) |
| 17 | \( 1 + 4147 T + p^{7} T^{2} \) |
| 19 | \( 1 + 3186 T + p^{7} T^{2} \) |
| 23 | \( 1 + 17784 T + p^{7} T^{2} \) |
| 29 | \( 1 + 3218 p T + p^{7} T^{2} \) |
| 31 | \( 1 + 124484 T + p^{7} T^{2} \) |
| 37 | \( 1 - 273661 T + p^{7} T^{2} \) |
| 41 | \( 1 - 585816 T + p^{7} T^{2} \) |
| 43 | \( 1 + 533559 T + p^{7} T^{2} \) |
| 47 | \( 1 + 530055 T + p^{7} T^{2} \) |
| 53 | \( 1 + 615288 T + p^{7} T^{2} \) |
| 59 | \( 1 + 392514 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1878064 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3971438 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3746601 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2485802 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1264456 T + p^{7} T^{2} \) |
| 83 | \( 1 - 434308 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5830810 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2045330 T + p^{7} T^{2} \) |
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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
−17.76836452100041457167108407354, −16.06553551976347966535830681578, −14.89504920411477867655404477823, −13.03012169578802712378885095075, −11.75480610596187868227619493229, −10.90779111678651162952004190529, −7.903338672769756218426902304095, −5.51947618671758194178021307214, −4.49862629470829445104686881139, 0,
4.49862629470829445104686881139, 5.51947618671758194178021307214, 7.903338672769756218426902304095, 10.90779111678651162952004190529, 11.75480610596187868227619493229, 13.03012169578802712378885095075, 14.89504920411477867655404477823, 16.06553551976347966535830681578, 17.76836452100041457167108407354
