| L(s) = 1 | + 3-s + 2·5-s + 7-s + 2·11-s − 5·13-s + 2·15-s − 6·17-s − 8·19-s + 21-s + 2·23-s + 3·25-s − 27-s + 8·29-s + 14·31-s + 2·33-s + 2·35-s + 2·37-s − 5·39-s − 6·41-s + 43-s + 16·47-s + 7·49-s − 6·51-s + 8·53-s + 4·55-s − 8·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 0.603·11-s − 1.38·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s + 0.218·21-s + 0.417·23-s + 3/5·25-s − 0.192·27-s + 1.48·29-s + 2.51·31-s + 0.348·33-s + 0.338·35-s + 0.328·37-s − 0.800·39-s − 0.937·41-s + 0.152·43-s + 2.33·47-s + 49-s − 0.840·51-s + 1.09·53-s + 0.539·55-s − 1.05·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.135017154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.135017154\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 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
−9.774071953234268133836787923825, −9.017351782105235704661329102356, −8.762533799290497301189910435392, −8.694909455872415211498391996161, −8.227983668458154878679779810771, −7.57694567422370405812528573273, −7.27194222144413256872752713044, −6.71745371907681729793230575128, −6.34709714391477244619975064562, −6.26179283286632056649863867784, −5.50149830537260337172591992413, −4.98223804556460384959442334211, −4.51075663052459725970132568446, −4.34440134627917175501263741264, −3.80785134740606867512962751691, −2.71145355158156269348305779803, −2.59633782652989730424763130363, −2.30287717376245722780033943106, −1.52103157847760467804321118471, −0.66506089601658386967343190241,
0.66506089601658386967343190241, 1.52103157847760467804321118471, 2.30287717376245722780033943106, 2.59633782652989730424763130363, 2.71145355158156269348305779803, 3.80785134740606867512962751691, 4.34440134627917175501263741264, 4.51075663052459725970132568446, 4.98223804556460384959442334211, 5.50149830537260337172591992413, 6.26179283286632056649863867784, 6.34709714391477244619975064562, 6.71745371907681729793230575128, 7.27194222144413256872752713044, 7.57694567422370405812528573273, 8.227983668458154878679779810771, 8.694909455872415211498391996161, 8.762533799290497301189910435392, 9.017351782105235704661329102356, 9.774071953234268133836787923825
