| L(s) = 1 | + 4-s − 2·5-s + 4·7-s + 9-s − 4·13-s + 16-s − 2·20-s + 4·23-s + 3·25-s + 4·28-s + 6·29-s − 8·35-s + 36-s − 2·45-s + 2·49-s − 4·52-s − 4·53-s + 8·59-s + 4·63-s + 64-s + 8·65-s − 4·67-s − 8·71-s − 2·80-s + 81-s + 12·83-s − 16·91-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.10·13-s + 1/4·16-s − 0.447·20-s + 0.834·23-s + 3/5·25-s + 0.755·28-s + 1.11·29-s − 1.35·35-s + 1/6·36-s − 0.298·45-s + 2/7·49-s − 0.554·52-s − 0.549·53-s + 1.04·59-s + 0.503·63-s + 1/8·64-s + 0.992·65-s − 0.488·67-s − 0.949·71-s − 0.223·80-s + 1/9·81-s + 1.31·83-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299555419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299555419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−8.255440688702926983790862388539, −7.69767636441248496320730664975, −7.50086921716889402509496991409, −7.10246483644061778149404136209, −6.60238351869316058908463659641, −6.11317224193381534591427075812, −5.33667024240828010769032910917, −4.98087906222688115423223689690, −4.59694498611125105823954123005, −4.23822265356238854864224738186, −3.42855656493158092255591716389, −2.93168194310820294528279776060, −2.23765501285652080229944622064, −1.62037155346921622709132301845, −0.76672817479135171126646777636,
0.76672817479135171126646777636, 1.62037155346921622709132301845, 2.23765501285652080229944622064, 2.93168194310820294528279776060, 3.42855656493158092255591716389, 4.23822265356238854864224738186, 4.59694498611125105823954123005, 4.98087906222688115423223689690, 5.33667024240828010769032910917, 6.11317224193381534591427075812, 6.60238351869316058908463659641, 7.10246483644061778149404136209, 7.50086921716889402509496991409, 7.69767636441248496320730664975, 8.255440688702926983790862388539
