| L(s) = 1 | + 2·3-s − 4·5-s + 2·9-s + 2·11-s + 4·13-s − 8·15-s + 8·17-s + 10·19-s + 8·25-s + 6·27-s − 12·29-s + 16·31-s + 4·33-s + 4·37-s + 8·39-s + 6·43-s − 8·45-s − 2·49-s + 16·51-s − 4·53-s − 8·55-s + 20·57-s − 6·59-s + 12·61-s − 16·65-s + 6·67-s + 16·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 2/3·9-s + 0.603·11-s + 1.10·13-s − 2.06·15-s + 1.94·17-s + 2.29·19-s + 8/5·25-s + 1.15·27-s − 2.22·29-s + 2.87·31-s + 0.696·33-s + 0.657·37-s + 1.28·39-s + 0.914·43-s − 1.19·45-s − 2/7·49-s + 2.24·51-s − 0.549·53-s − 1.07·55-s + 2.64·57-s − 0.781·59-s + 1.53·61-s − 1.98·65-s + 0.733·67-s + 1.84·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.249609195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.249609195\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−9.954281070457685068289726245157, −9.526817285620139489378377456741, −9.409920478091233746259209882163, −8.900992957953042825183609476697, −8.187166983210122349020402465142, −8.095042735430679895525811124551, −7.74548334530815184597716841813, −7.61796878861334286093134699213, −6.82688046566606086392714406137, −6.60779301427640553246739078925, −5.80860091744354574857771233461, −5.38960505960331554720034584180, −4.83029353859977792929546078694, −4.15098899647516312542627175088, −3.77906452622817252751683716534, −3.38847437700588197707927242378, −3.13135572245064270892446742574, −2.47558021980315297150068030380, −1.09494183249180906219837722251, −1.07629744124903481907801786640,
1.07629744124903481907801786640, 1.09494183249180906219837722251, 2.47558021980315297150068030380, 3.13135572245064270892446742574, 3.38847437700588197707927242378, 3.77906452622817252751683716534, 4.15098899647516312542627175088, 4.83029353859977792929546078694, 5.38960505960331554720034584180, 5.80860091744354574857771233461, 6.60779301427640553246739078925, 6.82688046566606086392714406137, 7.61796878861334286093134699213, 7.74548334530815184597716841813, 8.095042735430679895525811124551, 8.187166983210122349020402465142, 8.900992957953042825183609476697, 9.409920478091233746259209882163, 9.526817285620139489378377456741, 9.954281070457685068289726245157
