| L(s) = 1 | − 3-s − 4·4-s + 5-s − 7-s − 2·9-s + 4·12-s − 10·13-s − 15-s + 12·16-s − 4·20-s + 21-s − 12·23-s + 25-s + 5·27-s + 4·28-s − 35-s + 8·36-s + 10·39-s + 24·41-s − 2·45-s − 12·48-s + 49-s + 40·52-s + 24·53-s + 4·60-s + 2·63-s − 32·64-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.15·12-s − 2.77·13-s − 0.258·15-s + 3·16-s − 0.894·20-s + 0.218·21-s − 2.50·23-s + 1/5·25-s + 0.962·27-s + 0.755·28-s − 0.169·35-s + 4/3·36-s + 1.60·39-s + 3.74·41-s − 0.298·45-s − 1.73·48-s + 1/7·49-s + 5.54·52-s + 3.29·53-s + 0.516·60-s + 0.251·63-s − 4·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 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
−8.722193961422815471879095064652, −7.80156257853413862136981277007, −7.73929750781885351517852687666, −7.24281290023688107365847358204, −6.30078072165756181986797640838, −5.95865885309513387304829184659, −5.36755822629614084385217196812, −5.30121167828360660438204787375, −4.48390865811726709341858080699, −4.25386150379293419498976865827, −3.67606993779998961598679741611, −2.61198254943438177491615875407, −2.35094220433354059590404276967, −0.75515587387361470267464831173, 0,
0.75515587387361470267464831173, 2.35094220433354059590404276967, 2.61198254943438177491615875407, 3.67606993779998961598679741611, 4.25386150379293419498976865827, 4.48390865811726709341858080699, 5.30121167828360660438204787375, 5.36755822629614084385217196812, 5.95865885309513387304829184659, 6.30078072165756181986797640838, 7.24281290023688107365847358204, 7.73929750781885351517852687666, 7.80156257853413862136981277007, 8.722193961422815471879095064652
