| L(s) = 1 | − 2·2-s + 5·5-s − 14·7-s + 8·8-s − 10·10-s + 3·11-s − 47·13-s + 28·14-s − 16·16-s + 78·17-s + 64·19-s − 6·22-s − 99·23-s + 94·26-s + 51·29-s − 83·31-s − 156·34-s − 70·35-s + 628·37-s − 128·38-s + 40·40-s − 108·41-s − 299·43-s + 198·46-s + 531·47-s + 343·49-s − 1.12e3·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.0822·11-s − 1.00·13-s + 0.534·14-s − 1/4·16-s + 1.11·17-s + 0.772·19-s − 0.0581·22-s − 0.897·23-s + 0.709·26-s + 0.326·29-s − 0.480·31-s − 0.786·34-s − 0.338·35-s + 2.79·37-s − 0.546·38-s + 0.158·40-s − 0.411·41-s − 1.06·43-s + 0.634·46-s + 1.64·47-s + 49-s − 2.92·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.994104088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994104088\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 p T - 3 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 1322 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 47 T + 12 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 39 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 99 T - 2366 T^{2} + 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 51 T - 21788 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 83 T - 22902 T^{2} + 83 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 314 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 108 T - 57257 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 299 T + 9894 T^{2} + 299 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 531 T + 178138 T^{2} - 531 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 564 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T - 205235 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 230 T - 174081 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 p T - 51 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1106 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 739 T + 53082 T^{2} - 739 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1086 T + 607609 T^{2} - 1086 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1642 T + 1783491 T^{2} - 1642 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.845870691611995717984641458304, −9.609625312000271720866219396771, −9.392613911383855732295191547406, −9.079844379451559881394285467200, −8.151929858906485260219286291636, −8.088330358899608549016893379439, −7.43440491831584816413119921765, −7.38905746700525049946178355642, −6.43177262886921086682537542016, −6.31633808031043205313119773919, −5.76130576436338695690682934019, −5.20208821661420589669551789217, −4.75173724408871640606876172233, −4.21734633881263796552916328204, −3.31367053725196519427867388262, −3.26015183677310935929684309146, −2.25748152377766990889966073700, −1.94655260279876410072116731754, −0.835317840770469371927677103767, −0.58679236791753662452789469830,
0.58679236791753662452789469830, 0.835317840770469371927677103767, 1.94655260279876410072116731754, 2.25748152377766990889966073700, 3.26015183677310935929684309146, 3.31367053725196519427867388262, 4.21734633881263796552916328204, 4.75173724408871640606876172233, 5.20208821661420589669551789217, 5.76130576436338695690682934019, 6.31633808031043205313119773919, 6.43177262886921086682537542016, 7.38905746700525049946178355642, 7.43440491831584816413119921765, 8.088330358899608549016893379439, 8.151929858906485260219286291636, 9.079844379451559881394285467200, 9.392613911383855732295191547406, 9.609625312000271720866219396771, 9.845870691611995717984641458304
