| L(s) = 1 | + 8·5-s + 20·7-s − 3·9-s − 20·11-s + 20·17-s − 38·19-s + 40·23-s − 2·25-s + 160·35-s + 20·43-s − 24·45-s + 160·47-s + 202·49-s − 160·55-s − 20·61-s − 60·63-s − 20·73-s − 400·77-s + 9·81-s − 140·83-s + 160·85-s − 304·95-s + 60·99-s + 320·115-s + 400·119-s + 58·121-s − 344·125-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 20/7·7-s − 1/3·9-s − 1.81·11-s + 1.17·17-s − 2·19-s + 1.73·23-s − 0.0799·25-s + 32/7·35-s + 0.465·43-s − 0.533·45-s + 3.40·47-s + 4.12·49-s − 2.90·55-s − 0.327·61-s − 0.952·63-s − 0.273·73-s − 5.19·77-s + 1/9·81-s − 1.68·83-s + 1.88·85-s − 3.19·95-s + 0.606·99-s + 2.78·115-s + 3.36·119-s + 0.479·121-s − 2.75·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.179174188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.179174188\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 250 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 482 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1622 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2162 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3890 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5762 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 718 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12182 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 5042 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13010 T^{2} + p^{4} T^{4} \) |
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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
−10.24259394236443046104233393133, −9.825955110803486473240860427349, −9.132228358088393821893608003402, −8.688639601444591282864973441124, −8.584515737250789943519832300435, −7.941212978899379177460443938291, −7.62540078567229978575648815044, −7.42975830915162017074287945280, −6.66507306117713345129971012041, −5.91085941546642095568469417666, −5.54961374791959487081050150798, −5.49543040289886543915156586544, −4.89512405196761359331395626906, −4.52278658674846324482867718016, −4.00032308473241694158960517370, −2.87972752214784247179963544364, −2.43605058558670049919778434169, −2.00444441928150031958455513146, −1.54978647183617608738700598063, −0.74183601483777239773206508222,
0.74183601483777239773206508222, 1.54978647183617608738700598063, 2.00444441928150031958455513146, 2.43605058558670049919778434169, 2.87972752214784247179963544364, 4.00032308473241694158960517370, 4.52278658674846324482867718016, 4.89512405196761359331395626906, 5.49543040289886543915156586544, 5.54961374791959487081050150798, 5.91085941546642095568469417666, 6.66507306117713345129971012041, 7.42975830915162017074287945280, 7.62540078567229978575648815044, 7.941212978899379177460443938291, 8.584515737250789943519832300435, 8.688639601444591282864973441124, 9.132228358088393821893608003402, 9.825955110803486473240860427349, 10.24259394236443046104233393133
