| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s − 2·9-s + 10-s + 12-s + 15-s − 16-s + 2·18-s + 20-s − 3·24-s + 25-s + 5·27-s − 30-s + 7·31-s − 5·32-s + 2·36-s − 9·37-s − 3·40-s + 3·41-s − 6·43-s + 2·45-s + 48-s + 11·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s + 0.258·15-s − 1/4·16-s + 0.471·18-s + 0.223·20-s − 0.612·24-s + 1/5·25-s + 0.962·27-s − 0.182·30-s + 1.25·31-s − 0.883·32-s + 1/3·36-s − 1.47·37-s − 0.474·40-s + 0.468·41-s − 0.914·43-s + 0.298·45-s + 0.144·48-s + 11/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4745373084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4745373084\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.135302092333353652652186528333, −8.568822291342378708563372922224, −8.194417309724579054838462347537, −7.77531817530388024228130269937, −7.11431484901848047337285972916, −6.84899310638551761424185074277, −6.01612086119737147435067066575, −5.69886437390822823949011200716, −4.99207371310814885896269263171, −4.63021633725152443209733084264, −4.01480313584461926637256837089, −3.31874751359050490312370733802, −2.61237471789384733489531182439, −1.53647528102741022128099740579, −0.52408110145612652103271752356,
0.52408110145612652103271752356, 1.53647528102741022128099740579, 2.61237471789384733489531182439, 3.31874751359050490312370733802, 4.01480313584461926637256837089, 4.63021633725152443209733084264, 4.99207371310814885896269263171, 5.69886437390822823949011200716, 6.01612086119737147435067066575, 6.84899310638551761424185074277, 7.11431484901848047337285972916, 7.77531817530388024228130269937, 8.194417309724579054838462347537, 8.568822291342378708563372922224, 9.135302092333353652652186528333
