| L(s) = 1 | + 2·5-s + 2·7-s − 4·9-s − 2·11-s − 4·13-s − 4·17-s + 4·23-s + 3·25-s + 4·29-s − 4·31-s + 4·35-s − 16·37-s − 8·41-s − 4·43-s − 8·45-s + 8·47-s + 3·49-s − 8·53-s − 4·55-s − 12·59-s − 8·61-s − 8·63-s − 8·65-s + 20·67-s + 8·71-s + 4·73-s − 4·77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 4/3·9-s − 0.603·11-s − 1.10·13-s − 0.970·17-s + 0.834·23-s + 3/5·25-s + 0.742·29-s − 0.718·31-s + 0.676·35-s − 2.63·37-s − 1.24·41-s − 0.609·43-s − 1.19·45-s + 1.16·47-s + 3/7·49-s − 1.09·53-s − 0.539·55-s − 1.56·59-s − 1.02·61-s − 1.00·63-s − 0.992·65-s + 2.44·67-s + 0.949·71-s + 0.468·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 130 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 136 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 202 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 148 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.915968109373235628863629679772, −7.57346891403331798425489400831, −7.06780579430732611397843844202, −6.87444455295250154815358869750, −6.34073410783041703955721384159, −6.31501791307017086210691254263, −5.43012751957745481668934397366, −5.38408291468749085010048374366, −5.04292068450971163102680142626, −4.96051397792433362376344005370, −4.32134159202759200029868788480, −3.82100203955440285883365149998, −3.20814095249038650711534792826, −3.00215027472201759014405354782, −2.38541174753680098565852557914, −2.24707492110363091641160177226, −1.66823027378942127351181292679, −1.18486306905568471172342482872, 0, 0,
1.18486306905568471172342482872, 1.66823027378942127351181292679, 2.24707492110363091641160177226, 2.38541174753680098565852557914, 3.00215027472201759014405354782, 3.20814095249038650711534792826, 3.82100203955440285883365149998, 4.32134159202759200029868788480, 4.96051397792433362376344005370, 5.04292068450971163102680142626, 5.38408291468749085010048374366, 5.43012751957745481668934397366, 6.31501791307017086210691254263, 6.34073410783041703955721384159, 6.87444455295250154815358869750, 7.06780579430732611397843844202, 7.57346891403331798425489400831, 7.915968109373235628863629679772
