| L(s) = 1 | + 3·2-s + 4·4-s + 3·8-s + 3·16-s + 12·17-s − 4·19-s + 6·23-s − 8·31-s + 6·32-s + 36·34-s − 12·38-s + 18·46-s − 14·49-s + 24·53-s − 2·61-s − 24·62-s + 5·64-s + 48·68-s − 16·76-s − 16·79-s + 6·83-s + 24·92-s − 42·98-s + 72·106-s + 14·109-s − 22·121-s − 6·122-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 1.06·8-s + 3/4·16-s + 2.91·17-s − 0.917·19-s + 1.25·23-s − 1.43·31-s + 1.06·32-s + 6.17·34-s − 1.94·38-s + 2.65·46-s − 2·49-s + 3.29·53-s − 0.256·61-s − 3.04·62-s + 5/8·64-s + 5.82·68-s − 1.83·76-s − 1.80·79-s + 0.658·83-s + 2.50·92-s − 4.24·98-s + 6.99·106-s + 1.34·109-s − 2·121-s − 0.543·122-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.300057667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.300057667\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 24 T + 245 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−10.78038521942540907100259045443, −10.35512206999629643908977223765, −10.03621504316639370617559622912, −9.567299117901375406968753704114, −8.914917417765390268653735458889, −8.558508402508444795291817557582, −7.87665575801581715244869725539, −7.54860608131123895356374003622, −7.09471470718208076698673689061, −6.53137444575247663719978475609, −5.92123140089364848706671948637, −5.55479613634857707357429754257, −5.25941550531456978974929114273, −4.87442971353380959391737872935, −4.08807451385268501567567573099, −3.87692393850453507716821001926, −3.08045219779734005587003432611, −3.07858435171798324011693354819, −1.92860018740434397351518724248, −1.05639703435404280964920717849,
1.05639703435404280964920717849, 1.92860018740434397351518724248, 3.07858435171798324011693354819, 3.08045219779734005587003432611, 3.87692393850453507716821001926, 4.08807451385268501567567573099, 4.87442971353380959391737872935, 5.25941550531456978974929114273, 5.55479613634857707357429754257, 5.92123140089364848706671948637, 6.53137444575247663719978475609, 7.09471470718208076698673689061, 7.54860608131123895356374003622, 7.87665575801581715244869725539, 8.558508402508444795291817557582, 8.914917417765390268653735458889, 9.567299117901375406968753704114, 10.03621504316639370617559622912, 10.35512206999629643908977223765, 10.78038521942540907100259045443
