L(s) = 1 | − 3-s + 9-s − 11-s − 5·13-s + 6·17-s − 19-s − 2·23-s − 27-s − 6·29-s − 4·31-s + 33-s + 37-s + 5·39-s + 6·41-s − 4·43-s − 6·47-s − 6·51-s + 57-s + 61-s + 5·67-s + 2·69-s − 6·71-s + 9·73-s − 5·79-s + 81-s + 6·83-s + 6·87-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 1.45·17-s − 0.229·19-s − 0.417·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.164·37-s + 0.800·39-s + 0.937·41-s − 0.609·43-s − 0.875·47-s − 0.840·51-s + 0.132·57-s + 0.128·61-s + 0.610·67-s + 0.240·69-s − 0.712·71-s + 1.05·73-s − 0.562·79-s + 1/9·81-s + 0.658·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2236688581\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2236688581\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−12.59679461925732, −12.09975189778642, −11.88743916208614, −11.11636084397283, −10.96143603717267, −10.28309591460740, −9.859929024927176, −9.572339261609824, −9.149095673459073, −8.318239502485443, −7.887339792796867, −7.588969100875125, −7.009519747361788, −6.655237479118848, −5.864158072166306, −5.583075595371981, −5.112072222201060, −4.705569229321496, −3.921601881534322, −3.645309720428489, −2.795151152093984, −2.412298059979684, −1.647398128862500, −1.130363935876410, −0.1340374673951215,
0.1340374673951215, 1.130363935876410, 1.647398128862500, 2.412298059979684, 2.795151152093984, 3.645309720428489, 3.921601881534322, 4.705569229321496, 5.112072222201060, 5.583075595371981, 5.864158072166306, 6.655237479118848, 7.009519747361788, 7.588969100875125, 7.887339792796867, 8.318239502485443, 9.149095673459073, 9.572339261609824, 9.859929024927176, 10.28309591460740, 10.96143603717267, 11.11636084397283, 11.88743916208614, 12.09975189778642, 12.59679461925732