L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s − 3·13-s − 17-s − 6·19-s + 2·21-s + 2·23-s − 27-s + 5·29-s − 4·31-s + 33-s + 2·37-s + 3·39-s + 12·41-s − 6·43-s + 9·47-s − 3·49-s + 51-s − 3·53-s + 6·57-s − 2·59-s + 4·61-s − 2·63-s + 5·67-s − 2·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s − 0.242·17-s − 1.37·19-s + 0.436·21-s + 0.417·23-s − 0.192·27-s + 0.928·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s + 0.480·39-s + 1.87·41-s − 0.914·43-s + 1.31·47-s − 3/7·49-s + 0.140·51-s − 0.412·53-s + 0.794·57-s − 0.260·59-s + 0.512·61-s − 0.251·63-s + 0.610·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.003242342\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003242342\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.79078940989127, −12.65558672738063, −12.12659021905198, −11.50182137796502, −11.08151982986844, −10.63397038480127, −10.17922083087093, −9.790942830886476, −9.216127599627060, −8.828416064495593, −8.230366474105186, −7.628958699384681, −7.198292262180301, −6.700933432405555, −6.165856130280333, −5.927369214633378, −5.095234913566527, −4.779261675050820, −4.196275168685054, −3.664184427745023, −2.941763687258724, −2.408326317878395, −1.928371867317989, −0.9325881307607866, −0.3415991677925382,
0.3415991677925382, 0.9325881307607866, 1.928371867317989, 2.408326317878395, 2.941763687258724, 3.664184427745023, 4.196275168685054, 4.779261675050820, 5.095234913566527, 5.927369214633378, 6.165856130280333, 6.700933432405555, 7.198292262180301, 7.628958699384681, 8.230366474105186, 8.828416064495593, 9.216127599627060, 9.790942830886476, 10.17922083087093, 10.63397038480127, 11.08151982986844, 11.50182137796502, 12.12659021905198, 12.65558672738063, 12.79078940989127