| L(s) = 1 | + 488·7-s + 5.45e3·13-s − 1.07e4·19-s + 2.72e4·25-s − 2.03e4·31-s − 1.30e5·37-s − 9.89e4·43-s − 5.66e4·49-s − 2.01e5·61-s − 8.71e5·67-s + 1.23e6·73-s − 1.02e6·79-s + 2.66e6·91-s + 8.54e4·97-s + 2.84e6·103-s − 2.80e5·109-s − 2.18e6·121-s + 127-s + 131-s − 5.26e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.42·7-s + 2.48·13-s − 1.57·19-s + 1.74·25-s − 0.682·31-s − 2.56·37-s − 1.24·43-s − 0.481·49-s − 0.886·61-s − 2.89·67-s + 3.18·73-s − 2.08·79-s + 3.53·91-s + 0.0935·97-s + 2.60·103-s − 0.216·109-s − 1.23·121-s − 2.23·133-s + 2.62·169-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
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.424960479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.424960479\) |
| \(L(4)\) |
|
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 | $C_2^2$ | \( 1 - 1088 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 244 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2182606 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2728 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 13547360 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5392 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 289848386 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 575602720 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10172 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 65006 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4637059040 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 49480 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 5600983390 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 43709554208 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 44614042990 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 100610 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 435736 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 252585347330 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 619568 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 514340 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 594384805586 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 810559043264 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 42704 T + p^{6} T^{2} )^{2} \) |
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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
−10.32773329545658011693920961446, −9.349833827529564094497672505671, −8.843683601304849558276621027775, −8.597948063879071472606167468873, −8.470513926861914069606038070186, −7.88762173173168926947715715414, −7.38488741161573351738677601320, −6.59035953323363400039147962061, −6.56324248450893176182266966907, −5.94266546226535715693695635307, −5.33359526899324197357017060374, −4.84065891859500813199145808143, −4.55001359835843280548304567743, −3.63037109784096921956711504300, −3.62332607885578900798584038410, −2.78508155267604458019707850690, −1.93044728166223983123999889294, −1.38228286952842917645895876582, −1.38098598805862620243671024183, −0.30515290282052080420495107216,
0.30515290282052080420495107216, 1.38098598805862620243671024183, 1.38228286952842917645895876582, 1.93044728166223983123999889294, 2.78508155267604458019707850690, 3.62332607885578900798584038410, 3.63037109784096921956711504300, 4.55001359835843280548304567743, 4.84065891859500813199145808143, 5.33359526899324197357017060374, 5.94266546226535715693695635307, 6.56324248450893176182266966907, 6.59035953323363400039147962061, 7.38488741161573351738677601320, 7.88762173173168926947715715414, 8.470513926861914069606038070186, 8.597948063879071472606167468873, 8.843683601304849558276621027775, 9.349833827529564094497672505671, 10.32773329545658011693920961446
