| L(s) = 1 | − 2-s + 2·4-s + 4·5-s + 4·7-s − 5·8-s − 4·10-s − 2·11-s + 2·13-s − 4·14-s + 5·16-s + 6·17-s − 4·19-s + 8·20-s + 2·22-s − 6·23-s + 5·25-s − 2·26-s + 8·28-s + 4·29-s − 3·31-s − 10·32-s − 6·34-s + 16·35-s − 3·37-s + 4·38-s − 20·40-s − 4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s + 1.78·5-s + 1.51·7-s − 1.76·8-s − 1.26·10-s − 0.603·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s + 1.45·17-s − 0.917·19-s + 1.78·20-s + 0.426·22-s − 1.25·23-s + 25-s − 0.392·26-s + 1.51·28-s + 0.742·29-s − 0.538·31-s − 1.76·32-s − 1.02·34-s + 2.70·35-s − 0.493·37-s + 0.648·38-s − 3.16·40-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.484838456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.484838456\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
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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
−12.68522400706573042808288873370, −12.15426450409599511608426842844, −11.81025057505638265883708592369, −11.32705007789106230782260618041, −10.51661116892570110161586728721, −10.48024661562410809117275995165, −9.902330088778423068787203028341, −9.373838237260404718452307206033, −8.820695427120424241574013092030, −8.246535930921999168223185001496, −7.966233014049066905849841644330, −7.22406986068622315384115031300, −6.27570760253699139758996857516, −6.22333767532063700394517592893, −5.38175250394855804596159596545, −5.20866446729070686428160561998, −3.89375069261998848412767326432, −2.87627368937269559306127638353, −2.06956961744991906432622329790, −1.58396956880540069348832040896,
1.58396956880540069348832040896, 2.06956961744991906432622329790, 2.87627368937269559306127638353, 3.89375069261998848412767326432, 5.20866446729070686428160561998, 5.38175250394855804596159596545, 6.22333767532063700394517592893, 6.27570760253699139758996857516, 7.22406986068622315384115031300, 7.966233014049066905849841644330, 8.246535930921999168223185001496, 8.820695427120424241574013092030, 9.373838237260404718452307206033, 9.902330088778423068787203028341, 10.48024661562410809117275995165, 10.51661116892570110161586728721, 11.32705007789106230782260618041, 11.81025057505638265883708592369, 12.15426450409599511608426842844, 12.68522400706573042808288873370
