| L(s) = 1 | + 8·5-s + 30·7-s + 46·11-s + 92·13-s + 22·17-s + 86·19-s + 58·23-s − 25-s + 108·29-s − 136·31-s + 240·35-s − 180·37-s + 672·41-s + 70·43-s − 852·47-s + 190·49-s + 668·53-s + 368·55-s − 548·59-s + 284·61-s + 736·65-s − 1.16e3·67-s + 806·71-s − 1.51e3·73-s + 1.38e3·77-s + 22·79-s − 1.54e3·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s + 1.61·7-s + 1.26·11-s + 1.96·13-s + 0.313·17-s + 1.03·19-s + 0.525·23-s − 0.00799·25-s + 0.691·29-s − 0.787·31-s + 1.15·35-s − 0.799·37-s + 2.55·41-s + 0.248·43-s − 2.64·47-s + 0.553·49-s + 1.73·53-s + 0.902·55-s − 1.20·59-s + 0.596·61-s + 1.40·65-s − 2.11·67-s + 1.34·71-s − 2.42·73-s + 2.04·77-s + 0.0313·79-s − 2.03·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.555372333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.555372333\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 8 T + 13 p T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 30 T + 710 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 46 T + 1382 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 92 T + 6309 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 22 T - 2917 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 86 T + 10542 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 58 T + 24974 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 108 T + 51493 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 136 T + 12750 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 180 T + 109205 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 672 T + 249934 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 70 T + 53910 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 852 T + 388318 T^{2} + 852 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 668 T + 357854 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 548 T + 478598 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 284 T + 416037 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1162 T + 914766 T^{2} + 1162 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 806 T + 876422 T^{2} - 806 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1510 T + 1282935 T^{2} + 1510 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 648318 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1540 T + 1446230 T^{2} + 1540 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1323 T + p^{3} T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 472 T + 1378542 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.289233200526175469851439901761, −9.013674255585463779207573553266, −8.812866022693558867173122812110, −8.326845365386062085860053089879, −7.84817666555741214800580635581, −7.57985244683964226704062499901, −7.00095790654178333573182064760, −6.57726551890129183654280884917, −5.98642022016180531070181412972, −5.87699476666296666629555273519, −5.32093438937642539093137426793, −4.86132004068625778622628392931, −4.27287345149889360000851541292, −4.04620040276696125545287083227, −3.18509028931402519998479296769, −3.08442202252729186237901666631, −1.87396727198346019475732758871, −1.71262414105534083622082589402, −1.16278220563897731806188355359, −0.77189245405439956048615638478,
0.77189245405439956048615638478, 1.16278220563897731806188355359, 1.71262414105534083622082589402, 1.87396727198346019475732758871, 3.08442202252729186237901666631, 3.18509028931402519998479296769, 4.04620040276696125545287083227, 4.27287345149889360000851541292, 4.86132004068625778622628392931, 5.32093438937642539093137426793, 5.87699476666296666629555273519, 5.98642022016180531070181412972, 6.57726551890129183654280884917, 7.00095790654178333573182064760, 7.57985244683964226704062499901, 7.84817666555741214800580635581, 8.326845365386062085860053089879, 8.812866022693558867173122812110, 9.013674255585463779207573553266, 9.289233200526175469851439901761
