| L(s) = 1 | + 38·9-s − 72·11-s − 232·19-s − 396·29-s − 480·31-s + 884·41-s + 430·49-s − 696·59-s − 1.14e3·61-s − 336·71-s + 1.56e3·79-s + 715·81-s − 2.06e3·89-s − 2.73e3·99-s − 1.34e3·101-s − 2.09e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.63e3·169-s + ⋯ |
| L(s) = 1 | + 1.40·9-s − 1.97·11-s − 2.80·19-s − 2.53·29-s − 2.78·31-s + 3.36·41-s + 1.25·49-s − 1.53·59-s − 2.39·61-s − 0.561·71-s + 2.23·79-s + 0.980·81-s − 2.46·89-s − 2.77·99-s − 1.32·101-s − 1.83·109-s + 0.921·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.19·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3601099984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3601099984\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 430 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2274 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24078 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34742 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 442 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 53982 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 277590 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 348 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 570 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 122662 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 760078 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 784 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 825478 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1034 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1679422 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.01519000093621228869767833557, −10.74842479010463585524391195036, −10.42741841137821374936144048474, −9.615326803355300093298382623778, −9.214618929795528962268078027681, −9.033825950637740568978483396681, −8.157366222194835609390403232798, −7.62395416003051745965847050786, −7.55903615579422406769202031892, −7.01529776914329486189020125320, −6.28995597233913080738906795438, −5.63771339666127352643021166319, −5.50385737179412422580952269148, −4.49729992251982666665775615118, −4.23358726321136088660764765934, −3.72082623150178188014131171741, −2.69272099632326363505226845923, −2.11455972983285637505213112264, −1.62414781016633943760384181179, −0.18982559968191486246037740198,
0.18982559968191486246037740198, 1.62414781016633943760384181179, 2.11455972983285637505213112264, 2.69272099632326363505226845923, 3.72082623150178188014131171741, 4.23358726321136088660764765934, 4.49729992251982666665775615118, 5.50385737179412422580952269148, 5.63771339666127352643021166319, 6.28995597233913080738906795438, 7.01529776914329486189020125320, 7.55903615579422406769202031892, 7.62395416003051745965847050786, 8.157366222194835609390403232798, 9.033825950637740568978483396681, 9.214618929795528962268078027681, 9.615326803355300093298382623778, 10.42741841137821374936144048474, 10.74842479010463585524391195036, 11.01519000093621228869767833557
