L(s) = 1 | − 3-s + 5-s + 9-s + 2·13-s − 15-s + 6·17-s − 8·19-s + 25-s − 27-s − 6·29-s − 4·31-s + 10·37-s − 2·39-s + 6·41-s − 4·43-s + 45-s − 6·51-s + 6·53-s + 8·57-s + 12·59-s − 10·61-s + 2·65-s − 4·67-s − 12·71-s + 10·73-s − 75-s − 8·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 1.64·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.840·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.886858607\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886858607\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.71255790774126, −14.20280687141894, −13.32156467875001, −13.03314449810014, −12.67885713248348, −11.98101319339784, −11.47853600936032, −10.90996660398930, −10.52577390229715, −9.944118315716642, −9.476602520026193, −8.795829380901138, −8.339583782811815, −7.548853412518320, −7.227568973120985, −6.287213434554290, −6.020559162066488, −5.559061119662327, −4.830425887845511, −4.142259969927957, −3.664421837666365, −2.790562818768234, −2.037439524964905, −1.367112172762159, −0.5242706823290154,
0.5242706823290154, 1.367112172762159, 2.037439524964905, 2.790562818768234, 3.664421837666365, 4.142259969927957, 4.830425887845511, 5.559061119662327, 6.020559162066488, 6.287213434554290, 7.227568973120985, 7.548853412518320, 8.339583782811815, 8.795829380901138, 9.476602520026193, 9.944118315716642, 10.52577390229715, 10.90996660398930, 11.47853600936032, 11.98101319339784, 12.67885713248348, 13.03314449810014, 13.32156467875001, 14.20280687141894, 14.71255790774126