| L(s) = 1 | − 180·5-s − 3.52e3·13-s + 3.27e3·17-s − 6.95e3·25-s − 3.20e4·29-s − 1.22e5·37-s + 1.97e5·41-s + 2.00e5·49-s − 5.50e5·53-s − 2.13e5·61-s + 6.34e5·65-s + 2.01e5·73-s − 5.89e5·85-s + 1.63e6·89-s + 1.11e6·97-s + 3.67e6·101-s + 2.92e6·109-s − 1.12e6·113-s + 7.08e5·121-s + 5.52e6·125-s + 127-s + 131-s + 137-s + 139-s + 5.76e6·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.43·5-s − 1.60·13-s + 0.666·17-s − 0.444·25-s − 1.31·29-s − 2.41·37-s + 2.85·41-s + 1.70·49-s − 3.69·53-s − 0.939·61-s + 2.30·65-s + 0.519·73-s − 0.960·85-s + 2.32·89-s + 1.22·97-s + 3.56·101-s + 2.25·109-s − 0.781·113-s + 0.400·121-s + 2.82·125-s + 1.88·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.504548093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504548093\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 18 p T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 200306 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 708770 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 1762 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 1638 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 62987326 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 114673250 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 16002 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1522190162 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 61130 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 98550 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10222364450 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10288348414 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 275346 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 19735007330 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 106634 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 123394697854 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 250713262370 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 100978 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 480206360594 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 238112486210 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 819054 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 557026 T + p^{6} T^{2} )^{2} \) |
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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
−9.881993506706252445834070932041, −9.583789667177507791709227125451, −9.050998744457089498391622991744, −8.783782208953455429265478418491, −7.85052007855208202183538588579, −7.79480151893774511742752731850, −7.35902889870774860689004807600, −7.28151579685485040361278825137, −6.21729008780054758789168589448, −6.05988833459417665153683482316, −5.23195840155517475240814165575, −4.90449013027666322127998256193, −4.35954226205795707575127700659, −3.83613236951036254265475318989, −3.40169462489702754875763135206, −2.92393029564909843533467596578, −2.04075702023201161055343467630, −1.77655396160485494189407163646, −0.53672440851527713158741673626, −0.45096262589917276772871538413,
0.45096262589917276772871538413, 0.53672440851527713158741673626, 1.77655396160485494189407163646, 2.04075702023201161055343467630, 2.92393029564909843533467596578, 3.40169462489702754875763135206, 3.83613236951036254265475318989, 4.35954226205795707575127700659, 4.90449013027666322127998256193, 5.23195840155517475240814165575, 6.05988833459417665153683482316, 6.21729008780054758789168589448, 7.28151579685485040361278825137, 7.35902889870774860689004807600, 7.79480151893774511742752731850, 7.85052007855208202183538588579, 8.783782208953455429265478418491, 9.050998744457089498391622991744, 9.583789667177507791709227125451, 9.881993506706252445834070932041
