| L(s) = 1 | − 3-s + 2·5-s + 2·7-s − 2·9-s − 2·11-s − 2·13-s − 2·15-s + 9·17-s − 9·19-s − 2·21-s − 16·23-s + 3·25-s + 2·27-s − 7·29-s − 5·31-s + 2·33-s + 4·35-s + 3·37-s + 2·39-s − 5·41-s − 4·43-s − 4·45-s − 2·47-s + 3·49-s − 9·51-s + 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s − 2/3·9-s − 0.603·11-s − 0.554·13-s − 0.516·15-s + 2.18·17-s − 2.06·19-s − 0.436·21-s − 3.33·23-s + 3/5·25-s + 0.384·27-s − 1.29·29-s − 0.898·31-s + 0.348·33-s + 0.676·35-s + 0.493·37-s + 0.320·39-s − 0.780·41-s − 0.609·43-s − 0.596·45-s − 0.291·47-s + 3/7·49-s − 1.26·51-s + 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13249600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13249600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 3 p T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 55 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 41 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 65 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 85 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T - 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 107 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 205 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.202291771525817068894475017954, −8.067464210458339668810120020108, −7.75544647959697324341432424762, −7.31479233866719182303397020267, −6.73912472484116101836529748356, −6.46321376564340113773256675754, −5.85538857997527870608778738646, −5.71966482278085944371779299034, −5.39380117047436735248037978195, −5.32770573608866530406381175695, −4.43453738527684569921546182870, −4.25637542501674371939580414143, −3.71462676586366903041159702243, −3.28428620990325479946114933008, −2.48758221826284606180735536989, −2.25525725598273649510170015073, −1.77580286047578491443949644123, −1.32561580890176224184356607069, 0, 0,
1.32561580890176224184356607069, 1.77580286047578491443949644123, 2.25525725598273649510170015073, 2.48758221826284606180735536989, 3.28428620990325479946114933008, 3.71462676586366903041159702243, 4.25637542501674371939580414143, 4.43453738527684569921546182870, 5.32770573608866530406381175695, 5.39380117047436735248037978195, 5.71966482278085944371779299034, 5.85538857997527870608778738646, 6.46321376564340113773256675754, 6.73912472484116101836529748356, 7.31479233866719182303397020267, 7.75544647959697324341432424762, 8.067464210458339668810120020108, 8.202291771525817068894475017954
