| L(s) = 1 | + 2·5-s − 4·11-s + 12·19-s − 25-s − 12·29-s + 20·31-s − 12·41-s − 49-s − 8·55-s − 12·61-s − 12·71-s − 8·79-s − 12·89-s + 24·95-s + 20·101-s − 20·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + 40·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s + 2.75·19-s − 1/5·25-s − 2.22·29-s + 3.59·31-s − 1.87·41-s − 1/7·49-s − 1.07·55-s − 1.53·61-s − 1.42·71-s − 0.900·79-s − 1.27·89-s + 2.46·95-s + 1.99·101-s − 1.91·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 3.21·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.396655016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396655016\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.385430690688612496799027988979, −8.663374052799159841748718653261, −8.310743389781576542629373279890, −7.918548963977376121674682815962, −7.58551599672267947532049211213, −7.28929553570732678245597481788, −6.85478291086975093072861791109, −6.22870801800108480643383373046, −6.06598709913296383658965695924, −5.38409126252294131123399073799, −5.37955324742581590718459697307, −4.91777452546150445392007757836, −4.43285638467264153634015248323, −3.83409296257433511446870684117, −3.21697733251261542419213825472, −2.84224052963291957307685786902, −2.63047523507144768482067838007, −1.64676607097372205083778305934, −1.47214282582009809343517324728, −0.51489169846173878162612166484,
0.51489169846173878162612166484, 1.47214282582009809343517324728, 1.64676607097372205083778305934, 2.63047523507144768482067838007, 2.84224052963291957307685786902, 3.21697733251261542419213825472, 3.83409296257433511446870684117, 4.43285638467264153634015248323, 4.91777452546150445392007757836, 5.37955324742581590718459697307, 5.38409126252294131123399073799, 6.06598709913296383658965695924, 6.22870801800108480643383373046, 6.85478291086975093072861791109, 7.28929553570732678245597481788, 7.58551599672267947532049211213, 7.918548963977376121674682815962, 8.310743389781576542629373279890, 8.663374052799159841748718653261, 9.385430690688612496799027988979
