| L(s) = 1 | + 2·3-s − 4-s − 2·5-s − 4·7-s + 3·9-s − 2·12-s − 4·13-s − 4·15-s − 3·16-s − 4·19-s + 2·20-s − 8·21-s + 3·25-s + 4·27-s + 4·28-s − 8·31-s + 8·35-s − 3·36-s + 4·37-s − 8·39-s − 4·43-s − 6·45-s − 6·48-s − 2·49-s + 4·52-s − 12·53-s − 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 9-s − 0.577·12-s − 1.10·13-s − 1.03·15-s − 3/4·16-s − 0.917·19-s + 0.447·20-s − 1.74·21-s + 3/5·25-s + 0.769·27-s + 0.755·28-s − 1.43·31-s + 1.35·35-s − 1/2·36-s + 0.657·37-s − 1.28·39-s − 0.609·43-s − 0.894·45-s − 0.866·48-s − 2/7·49-s + 0.554·52-s − 1.64·53-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.012646344264965441331115386482, −8.852164758348461743985909706088, −8.163900095235817963358518439646, −8.013413489399351804721201627120, −7.64619824307247359787219502145, −6.99571487166376361457011261989, −6.79993068119665872589232644652, −6.57172900199190558672907389573, −5.90398600771120009926611340783, −5.31363866241500678720553305179, −4.72565192280137514755377836435, −4.51984147268459278099873628979, −3.78473576132666394055654531978, −3.73464995750232537730113765193, −3.10289076230485859240868530454, −2.64695855207896765958810260536, −2.22860000122942472953322067492, −1.40086716744547073554651145561, 0, 0,
1.40086716744547073554651145561, 2.22860000122942472953322067492, 2.64695855207896765958810260536, 3.10289076230485859240868530454, 3.73464995750232537730113765193, 3.78473576132666394055654531978, 4.51984147268459278099873628979, 4.72565192280137514755377836435, 5.31363866241500678720553305179, 5.90398600771120009926611340783, 6.57172900199190558672907389573, 6.79993068119665872589232644652, 6.99571487166376361457011261989, 7.64619824307247359787219502145, 8.013413489399351804721201627120, 8.163900095235817963358518439646, 8.852164758348461743985909706088, 9.012646344264965441331115386482
