| L(s) = 1 | − 6·3-s − 82·5-s + 98·7-s − 114·9-s + 340·11-s − 910·13-s + 492·15-s + 3.21e3·17-s − 674·19-s − 588·21-s + 1.10e3·23-s + 1.89e3·25-s + 126·27-s − 8.06e3·29-s + 6.21e3·31-s − 2.04e3·33-s − 8.03e3·35-s + 8.51e3·37-s + 5.46e3·39-s − 1.30e3·41-s − 1.00e4·43-s + 9.34e3·45-s + 1.27e4·47-s + 7.20e3·49-s − 1.92e4·51-s + 1.12e4·53-s − 2.78e4·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 1.46·5-s + 0.755·7-s − 0.469·9-s + 0.847·11-s − 1.49·13-s + 0.564·15-s + 2.69·17-s − 0.428·19-s − 0.290·21-s + 0.435·23-s + 0.607·25-s + 0.0332·27-s − 1.78·29-s + 1.16·31-s − 0.326·33-s − 1.10·35-s + 1.02·37-s + 0.574·39-s − 0.121·41-s − 0.825·43-s + 0.688·45-s + 0.841·47-s + 3/7·49-s − 1.03·51-s + 0.548·53-s − 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 p T + 50 p T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 82 T + 4826 T^{2} + 82 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 340 T + 338582 T^{2} - 340 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 70 p T + 2514 p^{2} T^{2} + 70 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3216 T + 5412958 T^{2} - 3216 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 674 T + 4367142 T^{2} + 674 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 48 p T + 495170 p T^{2} - 48 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8064 T + 52795702 T^{2} + 8064 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6212 T + 51691038 T^{2} - 6212 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8512 T + 104326950 T^{2} - 8512 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1304 T + 73546526 T^{2} + 1304 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10004 T + 99339510 T^{2} + 10004 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12748 T + 323438270 T^{2} - 12748 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11220 T + 664373806 T^{2} - 11220 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12018 T - 285266426 T^{2} + 12018 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 102738 T + 4326026138 T^{2} + 102738 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24136 T + 1542084918 T^{2} - 24136 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 89720 T + 4576356302 T^{2} + 89720 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 55588 T + 3902743302 T^{2} + 55588 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 48824 T + 5430110622 T^{2} + 48824 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 35782 T + 4098945062 T^{2} - 35782 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18300 T + 3539716918 T^{2} + 18300 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 69984 T + 18325482398 T^{2} + 69984 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.03773221407874887514480943074, −9.662997700123196625103277743056, −9.197714777179346623066876141888, −8.632133469694280567934644288930, −8.055293378492272352878430780646, −7.66961035238385614050140663434, −7.51149085407284730129050631782, −7.07858730710418962077680986477, −6.20612156606446715442394197321, −5.73927819829446774210839168078, −5.32593117966838988713421688666, −4.66289318613400213650773080573, −4.27576510633589839234681588359, −3.71896435759485326372570724066, −3.09740284118574835355016953005, −2.60946332620890914514148008616, −1.46051276429475999513353927990, −1.15939878677846291994100853080, 0, 0,
1.15939878677846291994100853080, 1.46051276429475999513353927990, 2.60946332620890914514148008616, 3.09740284118574835355016953005, 3.71896435759485326372570724066, 4.27576510633589839234681588359, 4.66289318613400213650773080573, 5.32593117966838988713421688666, 5.73927819829446774210839168078, 6.20612156606446715442394197321, 7.07858730710418962077680986477, 7.51149085407284730129050631782, 7.66961035238385614050140663434, 8.055293378492272352878430780646, 8.632133469694280567934644288930, 9.197714777179346623066876141888, 9.662997700123196625103277743056, 10.03773221407874887514480943074
