| L(s) = 1 | + 3-s + 7-s − 9-s − 4·11-s + 3·13-s − 7·17-s − 4·19-s + 21-s + 2·23-s − 8·29-s − 8·31-s − 4·33-s + 3·37-s + 3·39-s + 4·43-s + 5·47-s − 9·49-s − 7·51-s − 13·53-s − 4·57-s − 7·59-s − 10·61-s − 63-s + 19·67-s + 2·69-s − 10·71-s − 11·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 1/3·9-s − 1.20·11-s + 0.832·13-s − 1.69·17-s − 0.917·19-s + 0.218·21-s + 0.417·23-s − 1.48·29-s − 1.43·31-s − 0.696·33-s + 0.493·37-s + 0.480·39-s + 0.609·43-s + 0.729·47-s − 9/7·49-s − 0.980·51-s − 1.78·53-s − 0.529·57-s − 0.911·59-s − 1.28·61-s − 0.125·63-s + 2.32·67-s + 0.240·69-s − 1.18·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 126 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 19 T + 220 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 172 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 130 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 178 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.030416815781157959752584356548, −7.977217821520905099999363309142, −7.39354650327697566571191015463, −7.23667368674003826937941985311, −6.68976542427338450105956692599, −6.30802067948381416789183641287, −5.84967530213675521522322952175, −5.75368627639428514483384005596, −5.08264882773962702346377998581, −4.77077535909309278291272269609, −4.40203846995579791572239276957, −4.02077463706326672991960983930, −3.42245061045840514289501903833, −3.19472630821373086275988257918, −2.53398639010901402454692856359, −2.28413223652911904488728077064, −1.77644446224955025913741417599, −1.28608363067669696909857755640, 0, 0,
1.28608363067669696909857755640, 1.77644446224955025913741417599, 2.28413223652911904488728077064, 2.53398639010901402454692856359, 3.19472630821373086275988257918, 3.42245061045840514289501903833, 4.02077463706326672991960983930, 4.40203846995579791572239276957, 4.77077535909309278291272269609, 5.08264882773962702346377998581, 5.75368627639428514483384005596, 5.84967530213675521522322952175, 6.30802067948381416789183641287, 6.68976542427338450105956692599, 7.23667368674003826937941985311, 7.39354650327697566571191015463, 7.977217821520905099999363309142, 8.030416815781157959752584356548
