| L(s) = 1 | − 4-s − 8·11-s + 16-s + 4·19-s + 16·29-s + 8·31-s − 20·41-s + 8·44-s + 10·49-s − 24·59-s − 4·61-s − 64-s − 4·76-s + 16·79-s − 20·89-s − 40·101-s − 8·109-s − 16·116-s + 26·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.41·11-s + 1/4·16-s + 0.917·19-s + 2.97·29-s + 1.43·31-s − 3.12·41-s + 1.20·44-s + 10/7·49-s − 3.12·59-s − 0.512·61-s − 1/8·64-s − 0.458·76-s + 1.80·79-s − 2.11·89-s − 3.98·101-s − 0.766·109-s − 1.48·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34222500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2081687461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2081687461\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.207672857865422703948717488355, −8.051691390751707992494283534825, −7.81027865582617784190063839661, −7.16346118872494318304047449206, −6.87456431896450521290729338210, −6.61704181109130251357099700072, −6.05834599149989146360436572984, −5.73713270730013353725998647163, −5.32474704536153101986721066779, −4.89974354996072214345556864808, −4.76510364973316533840518335260, −4.55496172354194629201673658959, −3.78442418218677366242854209576, −3.38183559743528123590714628350, −2.85700208559530177837165812390, −2.69421407146585450526738880090, −2.37111631717647593122169012166, −1.28126572348128815560942206798, −1.23154781727422314673933464467, −0.12626232242913018570040288438,
0.12626232242913018570040288438, 1.23154781727422314673933464467, 1.28126572348128815560942206798, 2.37111631717647593122169012166, 2.69421407146585450526738880090, 2.85700208559530177837165812390, 3.38183559743528123590714628350, 3.78442418218677366242854209576, 4.55496172354194629201673658959, 4.76510364973316533840518335260, 4.89974354996072214345556864808, 5.32474704536153101986721066779, 5.73713270730013353725998647163, 6.05834599149989146360436572984, 6.61704181109130251357099700072, 6.87456431896450521290729338210, 7.16346118872494318304047449206, 7.81027865582617784190063839661, 8.051691390751707992494283534825, 8.207672857865422703948717488355
