| L(s) = 1 | − 4-s − 4·5-s − 8·11-s + 16-s + 4·20-s + 11·25-s − 6·29-s + 14·31-s − 14·41-s + 8·44-s − 49-s + 32·55-s + 10·59-s − 14·61-s − 64-s − 6·71-s − 24·79-s − 4·80-s − 28·89-s − 11·100-s + 20·101-s + 40·109-s + 6·116-s + 26·121-s − 14·124-s − 24·125-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s − 2.41·11-s + 1/4·16-s + 0.894·20-s + 11/5·25-s − 1.11·29-s + 2.51·31-s − 2.18·41-s + 1.20·44-s − 1/7·49-s + 4.31·55-s + 1.30·59-s − 1.79·61-s − 1/8·64-s − 0.712·71-s − 2.70·79-s − 0.447·80-s − 2.96·89-s − 1.09·100-s + 1.99·101-s + 3.83·109-s + 0.557·116-s + 2.36·121-s − 1.25·124-s − 2.14·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1194396059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1194396059\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.469463338640619199687141540940, −8.594913676883796212663754559519, −8.497004317098699165190103564565, −8.416117615896282663671511083726, −7.910722005995596124565416693415, −7.38012966793484340496372093760, −7.33180364803405527738611823975, −6.90369682292706166472455160857, −6.07955646815413868287928711214, −5.85631213248439403890025949048, −5.15915451911064044933536541852, −4.91890386073066649669256261273, −4.47193562158291097772231791017, −4.23418191196290084247982383671, −3.40334233731209527438055502086, −3.17439252291440844827159302477, −2.76297842385605601490845328848, −2.05925313190700804681672203670, −1.07333505384317772197202160200, −0.14666739278379121765149094181,
0.14666739278379121765149094181, 1.07333505384317772197202160200, 2.05925313190700804681672203670, 2.76297842385605601490845328848, 3.17439252291440844827159302477, 3.40334233731209527438055502086, 4.23418191196290084247982383671, 4.47193562158291097772231791017, 4.91890386073066649669256261273, 5.15915451911064044933536541852, 5.85631213248439403890025949048, 6.07955646815413868287928711214, 6.90369682292706166472455160857, 7.33180364803405527738611823975, 7.38012966793484340496372093760, 7.910722005995596124565416693415, 8.416117615896282663671511083726, 8.497004317098699165190103564565, 8.594913676883796212663754559519, 9.469463338640619199687141540940
