| L(s) = 1 | − 3-s − 10·5-s + 14·7-s + 9·9-s + 13·11-s + 19·13-s + 10·15-s − 29·17-s − 14·21-s + 75·25-s − 26·27-s − 23·29-s − 13·33-s − 140·35-s − 19·39-s − 90·45-s + 31·47-s + 147·49-s + 29·51-s − 130·55-s + 126·63-s − 190·65-s + 4·71-s − 68·73-s − 75·75-s + 182·77-s + 157·79-s + ⋯ |
| L(s) = 1 | − 1/3·3-s − 2·5-s + 2·7-s + 9-s + 1.18·11-s + 1.46·13-s + 2/3·15-s − 1.70·17-s − 2/3·21-s + 3·25-s − 0.962·27-s − 0.793·29-s − 0.393·33-s − 4·35-s − 0.487·39-s − 2·45-s + 0.659·47-s + 3·49-s + 0.568·51-s − 2.36·55-s + 2·63-s − 2.92·65-s + 4/71·71-s − 0.931·73-s − 75-s + 2.36·77-s + 1.98·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.584123478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584123478\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 8 T^{2} + p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T + 48 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 19 T + 192 T^{2} - 19 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 29 T + 552 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 T - 312 T^{2} + 23 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 31 T - 1248 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 157 T + 18408 T^{2} - 157 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 86 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 149 T + 12792 T^{2} + 149 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−13.10363644869881241316434155471, −12.53552795893333470010720545440, −11.80538352514309094138047065429, −11.72376702428500592678735399878, −11.10300462055168991650542521779, −10.97588770808163545031307232816, −10.51670340148121430264332502484, −9.310410198818516695115555952732, −8.768975734879259907951265117097, −8.514943582867642115925451320249, −7.79022872064364671092465970319, −7.43943685147355718148005473936, −6.83472427293603917267514991460, −6.20110184789952087784648578884, −5.12229019017352242287760409680, −4.47961396660048144887428600946, −4.08883127178518451143939924670, −3.66087341456665241523076123552, −1.90785720639598481407558956583, −0.962573358583385147744273048565,
0.962573358583385147744273048565, 1.90785720639598481407558956583, 3.66087341456665241523076123552, 4.08883127178518451143939924670, 4.47961396660048144887428600946, 5.12229019017352242287760409680, 6.20110184789952087784648578884, 6.83472427293603917267514991460, 7.43943685147355718148005473936, 7.79022872064364671092465970319, 8.514943582867642115925451320249, 8.768975734879259907951265117097, 9.310410198818516695115555952732, 10.51670340148121430264332502484, 10.97588770808163545031307232816, 11.10300462055168991650542521779, 11.72376702428500592678735399878, 11.80538352514309094138047065429, 12.53552795893333470010720545440, 13.10363644869881241316434155471
