| L(s) = 1 | + 8·5-s − 294·9-s − 376·13-s + 1.53e3·17-s − 6.20e3·25-s − 5.75e3·29-s + 3.06e4·37-s − 2.44e4·41-s − 2.35e3·45-s − 5.96e3·49-s + 6.68e4·53-s + 6.64e4·61-s − 3.00e3·65-s + 3.78e4·73-s + 2.73e4·81-s + 1.22e4·85-s + 1.55e5·89-s − 1.57e5·97-s + 3.07e4·101-s + 2.86e5·109-s − 1.16e5·113-s + 1.10e5·117-s − 3.00e4·121-s − 7.47e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.143·5-s − 1.20·9-s − 0.617·13-s + 1.28·17-s − 1.98·25-s − 1.27·29-s + 3.68·37-s − 2.26·41-s − 0.173·45-s − 0.354·49-s + 3.26·53-s + 2.28·61-s − 0.0883·65-s + 0.830·73-s + 0.463·81-s + 0.183·85-s + 2.08·89-s − 1.70·97-s + 0.300·101-s + 2.31·109-s − 0.857·113-s + 0.746·117-s − 0.186·121-s − 0.427·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.568736524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568736524\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 98 p T^{2} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 5966 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 30070 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 188 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 766 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4812230 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6651886 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2876 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 29445826 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 15348 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 12202 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 229650614 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 47177182 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 33412 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1322710870 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 33204 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 315180634 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2722589134 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 18906 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6148693790 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 386889146 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 77866 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 78930 T + p^{5} T^{2} )^{2} \) |
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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
−11.45723696146855747994066518683, −11.15745967944002847639549671874, −10.18674196911679839444084925759, −10.02354731887554543958255344906, −9.554772147276458631049035621618, −9.077631652437235948349423622543, −8.287021894329324939102413150620, −8.117657390668765888590289589038, −7.49012350068650010499785141094, −7.07189172344346773168118159152, −6.12960042052453110923575307020, −5.86515035430440069442132720583, −5.38447147487608977247564449549, −4.81776555954974623539863966690, −3.71142289874408256218449752864, −3.69069583285540749762935802944, −2.44410011346242500932853072838, −2.34367335000521348243510328136, −1.15986141624890361936710548836, −0.38042943745501175416701022218,
0.38042943745501175416701022218, 1.15986141624890361936710548836, 2.34367335000521348243510328136, 2.44410011346242500932853072838, 3.69069583285540749762935802944, 3.71142289874408256218449752864, 4.81776555954974623539863966690, 5.38447147487608977247564449549, 5.86515035430440069442132720583, 6.12960042052453110923575307020, 7.07189172344346773168118159152, 7.49012350068650010499785141094, 8.117657390668765888590289589038, 8.287021894329324939102413150620, 9.077631652437235948349423622543, 9.554772147276458631049035621618, 10.02354731887554543958255344906, 10.18674196911679839444084925759, 11.15745967944002847639549671874, 11.45723696146855747994066518683
