| L(s) = 1 | − 4-s + 5·9-s − 12·11-s + 16-s − 2·19-s − 18·29-s − 8·31-s − 5·36-s + 12·44-s + 13·49-s − 18·59-s − 20·61-s − 64-s − 12·71-s + 2·76-s + 20·79-s + 16·81-s + 24·89-s − 60·99-s + 36·101-s − 22·109-s + 18·116-s + 86·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s − 3.61·11-s + 1/4·16-s − 0.458·19-s − 3.34·29-s − 1.43·31-s − 5/6·36-s + 1.80·44-s + 13/7·49-s − 2.34·59-s − 2.56·61-s − 1/8·64-s − 1.42·71-s + 0.229·76-s + 2.25·79-s + 16/9·81-s + 2.54·89-s − 6.03·99-s + 3.58·101-s − 2.10·109-s + 1.67·116-s + 7.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5785254096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5785254096\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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
−10.42424864242956759141114593136, −9.876017528067997263728010117759, −9.363353566030147453465344299148, −9.139184014635326537383955715586, −8.660162121413821916064949680429, −7.77612921283094186400315734487, −7.72984663965283997999488472376, −7.36968635601476475027561036474, −7.36655446060614346946881406783, −6.18205371221264052512287134121, −5.94588866048212482367291290365, −5.20581549300400466359991490832, −5.15246677623423518985594486824, −4.60657500121310258618406628634, −4.01447318220950203180659533846, −3.48119750726470909916394202472, −2.89482614234160538226307188203, −2.04124522009189983557643337644, −1.83976765163521167928633654641, −0.33900283762510085792816425473,
0.33900283762510085792816425473, 1.83976765163521167928633654641, 2.04124522009189983557643337644, 2.89482614234160538226307188203, 3.48119750726470909916394202472, 4.01447318220950203180659533846, 4.60657500121310258618406628634, 5.15246677623423518985594486824, 5.20581549300400466359991490832, 5.94588866048212482367291290365, 6.18205371221264052512287134121, 7.36655446060614346946881406783, 7.36968635601476475027561036474, 7.72984663965283997999488472376, 7.77612921283094186400315734487, 8.660162121413821916064949680429, 9.139184014635326537383955715586, 9.363353566030147453465344299148, 9.876017528067997263728010117759, 10.42424864242956759141114593136
