| L(s) = 1 | − 3-s + 9-s − 11-s + 5·13-s + 6·17-s − 4·23-s − 27-s − 4·29-s + 7·31-s + 33-s − 4·37-s − 5·39-s − 4·41-s + 43-s + 6·47-s − 6·51-s + 2·53-s + 15·59-s + 4·61-s − 2·67-s + 4·69-s − 5·71-s − 9·73-s − 4·79-s + 81-s + 3·83-s + 4·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s + 1.38·13-s + 1.45·17-s − 0.834·23-s − 0.192·27-s − 0.742·29-s + 1.25·31-s + 0.174·33-s − 0.657·37-s − 0.800·39-s − 0.624·41-s + 0.152·43-s + 0.875·47-s − 0.840·51-s + 0.274·53-s + 1.95·59-s + 0.512·61-s − 0.244·67-s + 0.481·69-s − 0.593·71-s − 1.05·73-s − 0.450·79-s + 1/9·81-s + 0.329·83-s + 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.241261487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241261487\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.18979475399674, −12.84973349090642, −12.20139012710303, −11.78421085279533, −11.52656481391849, −10.85756140990461, −10.33257052179342, −10.12687621481281, −9.574907736478826, −8.846329843866050, −8.436818035582190, −7.976225196113944, −7.426309850172048, −6.905209538007127, −6.310709633619553, −5.784226960624597, −5.534273088543395, −4.928338238422004, −4.144831148537451, −3.784548516512798, −3.212873809468549, −2.507227661943708, −1.701124726480872, −1.147409486720077, −0.5038734449464854,
0.5038734449464854, 1.147409486720077, 1.701124726480872, 2.507227661943708, 3.212873809468549, 3.784548516512798, 4.144831148537451, 4.928338238422004, 5.534273088543395, 5.784226960624597, 6.310709633619553, 6.905209538007127, 7.426309850172048, 7.976225196113944, 8.436818035582190, 8.846329843866050, 9.574907736478826, 10.12687621481281, 10.33257052179342, 10.85756140990461, 11.52656481391849, 11.78421085279533, 12.20139012710303, 12.84973349090642, 13.18979475399674
