L(s) = 1 | + 2-s − 4-s − 3·8-s − 2·9-s − 4·13-s − 16-s + 2·17-s − 2·18-s + 8·19-s − 6·25-s − 4·26-s + 5·32-s + 2·34-s + 2·36-s + 8·38-s + 8·47-s − 2·49-s − 6·50-s + 4·52-s − 4·53-s + 7·64-s − 8·67-s − 2·68-s + 6·72-s − 8·76-s − 5·81-s + 16·83-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 2/3·9-s − 1.10·13-s − 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.83·19-s − 6/5·25-s − 0.784·26-s + 0.883·32-s + 0.342·34-s + 1/3·36-s + 1.29·38-s + 1.16·47-s − 2/7·49-s − 0.848·50-s + 0.554·52-s − 0.549·53-s + 7/8·64-s − 0.977·67-s − 0.242·68-s + 0.707·72-s − 0.917·76-s − 5/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7694790950\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7694790950\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−13.29480901314820395116553055383, −12.47860606469799974443984036732, −11.93259226357823232140431724447, −11.72832068978073525243612985214, −10.75100662817001349923979099653, −9.789115370436604117705818298937, −9.552128918559945705841722441351, −8.781803539780993843603843763167, −7.86908856597964737620953720124, −7.36547148424824287699212207874, −6.17653615753153086734821876420, −5.47448849864809007342052544464, −4.90526824858536973933461518159, −3.77471359399535732815739676770, −2.82633709454958812603882709287,
2.82633709454958812603882709287, 3.77471359399535732815739676770, 4.90526824858536973933461518159, 5.47448849864809007342052544464, 6.17653615753153086734821876420, 7.36547148424824287699212207874, 7.86908856597964737620953720124, 8.781803539780993843603843763167, 9.552128918559945705841722441351, 9.789115370436604117705818298937, 10.75100662817001349923979099653, 11.72832068978073525243612985214, 11.93259226357823232140431724447, 12.47860606469799974443984036732, 13.29480901314820395116553055383