| L(s) = 1 | − 8·2-s − 8·3-s + 48·4-s + 64·6-s + 98·7-s − 256·8-s − 122·9-s − 6·11-s − 384·12-s − 692·13-s − 784·14-s + 1.28e3·16-s − 504·17-s + 976·18-s − 68·19-s − 784·21-s + 48·22-s − 1.65e3·23-s + 2.04e3·24-s + 5.53e3·26-s + 520·27-s + 4.70e3·28-s + 2.85e3·29-s + 3.84e3·31-s − 6.14e3·32-s + 48·33-s + 4.03e3·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.513·3-s + 3/2·4-s + 0.725·6-s + 0.755·7-s − 1.41·8-s − 0.502·9-s − 0.0149·11-s − 0.769·12-s − 1.13·13-s − 1.06·14-s + 5/4·16-s − 0.422·17-s + 0.710·18-s − 0.0432·19-s − 0.387·21-s + 0.0211·22-s − 0.650·23-s + 0.725·24-s + 1.60·26-s + 0.137·27-s + 1.13·28-s + 0.629·29-s + 0.718·31-s − 1.06·32-s + 0.00767·33-s + 0.598·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 8 T + 62 p T^{2} + 8 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 319267 T^{2} + 6 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 692 T + 791202 T^{2} + 692 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 504 T + 2891842 T^{2} + 504 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 68 T + 4031898 T^{2} + 68 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1650 T + 11775811 T^{2} + 1650 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2850 T + 18981307 T^{2} - 2850 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 p^{2} T + 60813030 T^{2} - 4 p^{7} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11342 T + 128518059 T^{2} + 11342 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15288 T + 255102214 T^{2} - 15288 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 854 T + 293718579 T^{2} + 854 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15060 T + 382701250 T^{2} + 15060 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18252 T + 899607598 T^{2} - 18252 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 38196 T + 1793760286 T^{2} - 38196 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15656 T + 1482426030 T^{2} + 15656 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 64898 T + 3422484651 T^{2} + 64898 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 40086 T + 2804890507 T^{2} + 40086 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 73360 T + 2653250742 T^{2} - 73360 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 35470 T + 6292312179 T^{2} - 35470 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1956 T + 4393860814 T^{2} + 1956 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 19356 T + 7173245338 T^{2} + 19356 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 227468 T + 30109963914 T^{2} + 227468 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.33711526897830866645047702770, −10.02946957969860746871554294162, −9.497879106460474700906934208241, −9.067314704831133062362139735974, −8.455710215287439833498373523310, −8.243669381931985321088628439167, −7.64372096954620399097940626343, −7.28620597237907357541720528576, −6.59867476533970152355486963791, −6.34820271868283375185870671960, −5.40681831547882218165023702127, −5.31731174942797126796885713625, −4.44678942731446812111127286805, −3.83635028420286544276062956129, −2.66040923522346258716907509602, −2.55633352028904637032941960992, −1.62726277951026715129555352840, −1.06705904140191035540692314601, 0, 0,
1.06705904140191035540692314601, 1.62726277951026715129555352840, 2.55633352028904637032941960992, 2.66040923522346258716907509602, 3.83635028420286544276062956129, 4.44678942731446812111127286805, 5.31731174942797126796885713625, 5.40681831547882218165023702127, 6.34820271868283375185870671960, 6.59867476533970152355486963791, 7.28620597237907357541720528576, 7.64372096954620399097940626343, 8.243669381931985321088628439167, 8.455710215287439833498373523310, 9.067314704831133062362139735974, 9.497879106460474700906934208241, 10.02946957969860746871554294162, 10.33711526897830866645047702770
