| L(s) = 1 | − 19·2-s + 54·3-s + 59·4-s − 34·5-s − 1.02e3·6-s − 166·7-s + 2.18e3·8-s + 2.18e3·9-s + 646·10-s − 2.66e3·11-s + 3.18e3·12-s − 1.26e4·13-s + 3.15e3·14-s − 1.83e3·15-s − 2.95e4·16-s − 6.23e4·17-s − 4.15e4·18-s − 3.79e4·19-s − 2.00e3·20-s − 8.96e3·21-s + 5.05e4·22-s − 9.06e4·23-s + 1.17e5·24-s − 9.14e4·25-s + 2.40e5·26-s + 7.87e4·27-s − 9.79e3·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.15·3-s + 0.460·4-s − 0.121·5-s − 1.93·6-s − 0.182·7-s + 1.50·8-s + 9-s + 0.204·10-s − 0.603·11-s + 0.532·12-s − 1.59·13-s + 0.307·14-s − 0.140·15-s − 1.80·16-s − 3.07·17-s − 1.67·18-s − 1.27·19-s − 0.0560·20-s − 0.211·21-s + 1.01·22-s − 1.55·23-s + 1.74·24-s − 1.17·25-s + 2.68·26-s + 0.769·27-s − 0.0843·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 19 T + 151 p T^{2} + 19 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 34 T + 92642 T^{2} + 34 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 166 T + 604542 T^{2} + 166 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12670 T + 131517642 T^{2} + 12670 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 62344 T + 1748474222 T^{2} + 62344 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 37980 T + 2116242326 T^{2} + 37980 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 90686 T + 13407542 p^{2} T^{2} + 90686 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 13992 T - 4336731434 T^{2} - 13992 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 245000 T + 56312134590 T^{2} - 245000 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 327852 T + 183339535550 T^{2} - 327852 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 275932 T + 66680638310 T^{2} + 275932 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 244104 T + 335871625670 T^{2} + 244104 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 536926 T + 837168473222 T^{2} - 536926 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1821882 T + 2746994724802 T^{2} + 1821882 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2502028 T + 4952639534534 T^{2} - 2502028 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2191098 T + 5681315972690 T^{2} + 2191098 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1674784 T + 3291529437702 T^{2} - 1674784 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 368310 T + 18165736320454 T^{2} + 368310 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3336604 T + 16427959744566 T^{2} + 3336604 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1682618 T + 16894191271566 T^{2} + 1682618 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8376504 T + 62394156052870 T^{2} + 8376504 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9027204 T + 72708018275350 T^{2} - 9027204 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16703552 T + 231348397156350 T^{2} + 16703552 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.98670010996660889436115801397, −14.31418353247003247009108414080, −13.41225503171957001612759743024, −13.40117344418563627834254471603, −12.54048428809181014578224203976, −11.55730407581481833230865477445, −10.65590899007078622791152420509, −10.02522521346487438102947149933, −9.639869872363248728376599193094, −9.027690352366922453364218339432, −8.220478729827320135936699329167, −8.184598487889616380011604297496, −7.22791556149533111685298092638, −6.39893834415907885731105240713, −4.49766386795423484287052771081, −4.36762764114964872283046156133, −2.49295564199612109405118807087, −1.95117230460532082621767608575, 0, 0,
1.95117230460532082621767608575, 2.49295564199612109405118807087, 4.36762764114964872283046156133, 4.49766386795423484287052771081, 6.39893834415907885731105240713, 7.22791556149533111685298092638, 8.184598487889616380011604297496, 8.220478729827320135936699329167, 9.027690352366922453364218339432, 9.639869872363248728376599193094, 10.02522521346487438102947149933, 10.65590899007078622791152420509, 11.55730407581481833230865477445, 12.54048428809181014578224203976, 13.40117344418563627834254471603, 13.41225503171957001612759743024, 14.31418353247003247009108414080, 14.98670010996660889436115801397
