| L(s) = 1 | + 3-s + 5-s − 7-s − 5·11-s + 15-s + 4·17-s − 8·19-s − 21-s − 4·23-s + 5·25-s − 27-s + 10·29-s + 3·31-s − 5·33-s − 35-s − 4·37-s + 4·43-s − 6·47-s − 6·49-s + 4·51-s − 9·53-s − 5·55-s − 8·57-s + 11·59-s − 6·61-s + 2·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 1.50·11-s + 0.258·15-s + 0.970·17-s − 1.83·19-s − 0.218·21-s − 0.834·23-s + 25-s − 0.192·27-s + 1.85·29-s + 0.538·31-s − 0.870·33-s − 0.169·35-s − 0.657·37-s + 0.609·43-s − 0.875·47-s − 6/7·49-s + 0.560·51-s − 1.23·53-s − 0.674·55-s − 1.05·57-s + 1.43·59-s − 0.768·61-s + 0.244·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.834961662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834961662\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.964144387205616760757576587111, −9.472059038176369575204979014707, −8.829965770867876428337453515507, −8.476345920140551803579788351928, −8.437734929599080839022445815197, −7.77071468299319691845252194904, −7.57881509584073805797410453542, −6.94406259940964775861221891901, −6.40802998149937015491503655251, −6.18426617767866734472035526609, −5.77843658959272649656695787938, −4.96146644002096163886826022258, −4.91613989710912956000428964026, −4.30562658207975681101140838961, −3.66397785011834080852943130608, −2.91412062726369648038873762917, −2.90866236358056530313661970435, −2.18329792223318467754764112735, −1.61463216684733902763759458720, −0.52085885221818334410213551925,
0.52085885221818334410213551925, 1.61463216684733902763759458720, 2.18329792223318467754764112735, 2.90866236358056530313661970435, 2.91412062726369648038873762917, 3.66397785011834080852943130608, 4.30562658207975681101140838961, 4.91613989710912956000428964026, 4.96146644002096163886826022258, 5.77843658959272649656695787938, 6.18426617767866734472035526609, 6.40802998149937015491503655251, 6.94406259940964775861221891901, 7.57881509584073805797410453542, 7.77071468299319691845252194904, 8.437734929599080839022445815197, 8.476345920140551803579788351928, 8.829965770867876428337453515507, 9.472059038176369575204979014707, 9.964144387205616760757576587111
