| L(s) = 1 | − 5-s + 3·7-s + 2·11-s + 5·13-s − 5·17-s + 7·19-s + 5·23-s − 25-s − 3·29-s + 7·31-s − 3·35-s + 6·37-s − 12·41-s + 8·43-s + 3·47-s + 49-s − 10·53-s − 2·55-s + 14·59-s − 61-s − 5·65-s + 4·67-s − 8·71-s − 7·73-s + 6·77-s − 7·79-s + 25·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.603·11-s + 1.38·13-s − 1.21·17-s + 1.60·19-s + 1.04·23-s − 1/5·25-s − 0.557·29-s + 1.25·31-s − 0.507·35-s + 0.986·37-s − 1.87·41-s + 1.21·43-s + 0.437·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s + 1.82·59-s − 0.128·61-s − 0.620·65-s + 0.488·67-s − 0.949·71-s − 0.819·73-s + 0.683·77-s − 0.787·79-s + 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.381781691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.381781691\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 44 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 88 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 105 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 162 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 25 T + 314 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.83943281408053258433283545658, −10.58578571309101544899384529323, −9.693888308866421029156850687573, −9.576843065504241184558598482492, −8.799904979851025540572241785058, −8.752998909016910310563883055469, −8.061958899966239295067836908081, −7.916179513960599328550941398993, −7.12303397034834639704116629023, −6.98208663927452500695709114485, −6.18205891008164914208109827458, −5.97116825900122923094083502182, −5.05650003332228128234271788280, −4.92209729755964021901152361854, −4.22163329341672737002141606332, −3.75261783253952068556634593975, −3.19220113097511283317562820081, −2.42441613490697030993622988545, −1.51854295860303128436601914260, −0.964340671969873056565929306747,
0.964340671969873056565929306747, 1.51854295860303128436601914260, 2.42441613490697030993622988545, 3.19220113097511283317562820081, 3.75261783253952068556634593975, 4.22163329341672737002141606332, 4.92209729755964021901152361854, 5.05650003332228128234271788280, 5.97116825900122923094083502182, 6.18205891008164914208109827458, 6.98208663927452500695709114485, 7.12303397034834639704116629023, 7.916179513960599328550941398993, 8.061958899966239295067836908081, 8.752998909016910310563883055469, 8.799904979851025540572241785058, 9.576843065504241184558598482492, 9.693888308866421029156850687573, 10.58578571309101544899384529323, 10.83943281408053258433283545658
