| L(s) = 1 | + 13·2-s + 71·4-s − 58·5-s − 146·7-s + 65·8-s − 754·10-s + 130·13-s − 1.89e3·14-s − 1.72e3·16-s − 728·17-s + 828·19-s − 4.11e3·20-s + 238·23-s − 2.90e3·25-s + 1.69e3·26-s − 1.03e4·28-s + 696·29-s − 1.04e4·31-s − 1.26e4·32-s − 9.46e3·34-s + 8.46e3·35-s − 1.90e3·37-s + 1.07e4·38-s − 3.77e3·40-s + 3.64e4·41-s − 9.76e3·43-s + 3.09e3·46-s + ⋯ |
| L(s) = 1 | + 2.29·2-s + 2.21·4-s − 1.03·5-s − 1.12·7-s + 0.359·8-s − 2.38·10-s + 0.213·13-s − 2.58·14-s − 1.68·16-s − 0.610·17-s + 0.526·19-s − 2.30·20-s + 0.0938·23-s − 0.928·25-s + 0.490·26-s − 2.49·28-s + 0.153·29-s − 1.95·31-s − 2.17·32-s − 1.40·34-s + 1.16·35-s − 0.229·37-s + 1.20·38-s − 0.372·40-s + 3.38·41-s − 0.805·43-s + 0.215·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 13 T + 49 p T^{2} - 13 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 58 T + 6266 T^{2} + 58 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 146 T + 7230 T^{2} + 146 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 p T + 701634 T^{2} - 10 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 728 T + 1830542 T^{2} + 728 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 828 T - 865114 T^{2} - 828 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 238 T + 11988422 T^{2} - 238 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 24 p T + 22778374 T^{2} - 24 p^{6} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10480 T + 63595902 T^{2} + 10480 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1908 T + 74176718 T^{2} + 1908 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 36484 T + 564482438 T^{2} - 36484 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9768 T + 237972854 T^{2} + 9768 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 43742 T + 935775830 T^{2} + 43742 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12174 T + 457325722 T^{2} - 12174 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2788 T + 1399448534 T^{2} - 2788 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 25302 T + 1815376826 T^{2} - 25302 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 40520 T + 1779236982 T^{2} + 40520 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 31386 T + 3698331094 T^{2} + 31386 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 46780 T + 4437463638 T^{2} - 46780 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16850 T + 5148027246 T^{2} - 16850 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 79440 T + 5477266486 T^{2} - 79440 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 54204 T + 6019532470 T^{2} - 54204 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 241568 T + 30950947518 T^{2} + 241568 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−8.931431708842331438958174433069, −8.566336900563722790498523357812, −7.76529879715556474446979775756, −7.70741204655303759286400657426, −7.03648685121059569048477840639, −6.67472491105618210030220843255, −6.06684158990375712193917929059, −5.99842973114875992593106506249, −5.35218966121275115720773581751, −5.04952876902203971211477089819, −4.44274179819487309532323432235, −4.08325485213920861295326343453, −3.70174532888551101061617380441, −3.46415303601865956017875080158, −2.98343429292211900766823019409, −2.44043715985697775727899793437, −1.80371116891261997279370554382, −0.848547270516492788747347271733, 0, 0,
0.848547270516492788747347271733, 1.80371116891261997279370554382, 2.44043715985697775727899793437, 2.98343429292211900766823019409, 3.46415303601865956017875080158, 3.70174532888551101061617380441, 4.08325485213920861295326343453, 4.44274179819487309532323432235, 5.04952876902203971211477089819, 5.35218966121275115720773581751, 5.99842973114875992593106506249, 6.06684158990375712193917929059, 6.67472491105618210030220843255, 7.03648685121059569048477840639, 7.70741204655303759286400657426, 7.76529879715556474446979775756, 8.566336900563722790498523357812, 8.931431708842331438958174433069
