| L(s) = 1 | − 16·7-s − 9·9-s + 36·17-s − 144·23-s − 74·25-s − 32·31-s − 180·41-s + 864·47-s − 494·49-s + 144·63-s + 720·71-s − 52·73-s + 1.02e3·79-s + 81·81-s + 1.26e3·89-s − 2.10e3·97-s − 16·103-s − 2.26e3·113-s − 576·119-s + 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 324·153-s + ⋯ |
| L(s) = 1 | − 0.863·7-s − 1/3·9-s + 0.513·17-s − 1.30·23-s − 0.591·25-s − 0.185·31-s − 0.685·41-s + 2.68·47-s − 1.44·49-s + 0.287·63-s + 1.20·71-s − 0.0833·73-s + 1.45·79-s + 1/9·81-s + 1.50·89-s − 2.20·97-s − 0.0153·103-s − 1.88·113-s − 0.443·119-s + 1.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.171·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9823256682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9823256682\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 74 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1366 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4294 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3718 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 5978 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 45290 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 57098 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 275878 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 491302 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 512 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 267770 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1054 T + p^{3} T^{2} )^{2} \) |
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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
−10.08881067324306421500029197481, −9.737884546594726063377782717581, −9.238811468744759972913469501715, −9.095109911960597086915410145036, −8.255384401163519501133556230548, −8.097568209121498821018862034662, −7.62279060452625006954183538401, −7.09377638524978727077694808192, −6.51312737586089496700968102969, −6.31591941622518621019637956563, −5.53859165242114754583245503607, −5.53549213916671942318093155972, −4.70651228336037168848334717424, −4.08148615729278025458022051626, −3.65672284550610119733189711989, −3.20113523901968485947521524927, −2.47795874944550601029461285231, −1.99966830368175073866053642340, −1.11729544705206604353791469849, −0.29027593025749778408846499592,
0.29027593025749778408846499592, 1.11729544705206604353791469849, 1.99966830368175073866053642340, 2.47795874944550601029461285231, 3.20113523901968485947521524927, 3.65672284550610119733189711989, 4.08148615729278025458022051626, 4.70651228336037168848334717424, 5.53549213916671942318093155972, 5.53859165242114754583245503607, 6.31591941622518621019637956563, 6.51312737586089496700968102969, 7.09377638524978727077694808192, 7.62279060452625006954183538401, 8.097568209121498821018862034662, 8.255384401163519501133556230548, 9.095109911960597086915410145036, 9.238811468744759972913469501715, 9.737884546594726063377782717581, 10.08881067324306421500029197481
