| L(s) = 1 | + 80·5-s + 160·11-s + 24·19-s + 3.27e3·25-s + 9.12e3·29-s − 688·31-s + 2.84e4·41-s + 1.79e4·49-s + 1.28e4·55-s + 7.60e4·59-s − 1.64e4·61-s + 9.69e4·71-s + 1.85e4·79-s − 4.86e4·89-s + 1.92e3·95-s − 2.10e5·101-s + 8.28e4·109-s − 3.02e5·121-s + 1.20e4·125-s + 127-s + 131-s + 137-s + 139-s + 7.29e5·145-s + 149-s + 151-s − 5.50e4·155-s + ⋯ |
| L(s) = 1 | + 1.43·5-s + 0.398·11-s + 0.0152·19-s + 1.04·25-s + 2.01·29-s − 0.128·31-s + 2.64·41-s + 1.07·49-s + 0.570·55-s + 2.84·59-s − 0.564·61-s + 2.28·71-s + 0.334·79-s − 0.650·89-s + 0.0218·95-s − 2.05·101-s + 0.668·109-s − 1.88·121-s + 0.0686·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 2.88·145-s + 3.69e−6·149-s + 3.56e−6·151-s − 0.184·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.666265060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.666265060\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 16 p T + p^{5} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 17998 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 80 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 602042 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2540814 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 12 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8749086 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4560 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 344 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 119558314 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 14240 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 66775178 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 142745586 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 84864886 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 38000 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8206 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2524788838 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 48480 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2340933586 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 9264 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6760658886 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 24320 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1515110462 T^{2} + p^{10} T^{4} \) |
| 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
−12.07367796741795029133552115982, −11.49353406032626824673228809992, −10.94736781371061494027060391980, −10.45624677069687431765481900585, −9.965768940438104500625352254883, −9.594456587427478983868552526716, −9.053417070806083120433221891367, −8.610290844469607963396679934722, −7.972213851543689589465888100099, −7.30904903336771331826035202470, −6.52086347672874051070813537218, −6.39789183050700820515667307099, −5.55606019748555247702478411566, −5.23173192909360231508550734281, −4.35626446000563198366405428020, −3.76481943805278164227286650819, −2.56509623474353642146251131365, −2.43059818381194239510965387159, −1.29828393778213581644091259787, −0.75695663963673858563771283106,
0.75695663963673858563771283106, 1.29828393778213581644091259787, 2.43059818381194239510965387159, 2.56509623474353642146251131365, 3.76481943805278164227286650819, 4.35626446000563198366405428020, 5.23173192909360231508550734281, 5.55606019748555247702478411566, 6.39789183050700820515667307099, 6.52086347672874051070813537218, 7.30904903336771331826035202470, 7.972213851543689589465888100099, 8.610290844469607963396679934722, 9.053417070806083120433221891367, 9.594456587427478983868552526716, 9.965768940438104500625352254883, 10.45624677069687431765481900585, 10.94736781371061494027060391980, 11.49353406032626824673228809992, 12.07367796741795029133552115982
