| L(s) = 1 | − 2·3-s + 5-s + 3·7-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 3·17-s − 2·19-s − 6·21-s + 23-s + 25-s + 4·27-s − 7·31-s + 8·33-s + 3·35-s + 10·37-s + 4·39-s + 2·41-s + 2·43-s + 45-s + 7·47-s + 2·49-s + 6·51-s + 2·53-s − 4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.727·17-s − 0.458·19-s − 1.30·21-s + 0.208·23-s + 1/5·25-s + 0.769·27-s − 1.25·31-s + 1.39·33-s + 0.507·35-s + 1.64·37-s + 0.640·39-s + 0.312·41-s + 0.304·43-s + 0.149·45-s + 1.02·47-s + 2/7·49-s + 0.840·51-s + 0.274·53-s − 0.539·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.036954939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036954939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.12184126788774, −14.35565408538129, −14.11256618731802, −13.23637928801482, −12.83134403418036, −12.41438859839510, −11.66270341462914, −11.18991862828929, −10.85659713247335, −10.46114146116302, −9.755284676858554, −9.093052244157227, −8.499426667409493, −7.817251857634412, −7.409576898559139, −6.665511654217299, −6.010701258569859, −5.536933668953543, −4.966514721687253, −4.661939659377428, −3.852408798052976, −2.608133670226866, −2.312211475443340, −1.344039258992795, −0.4219273945731289,
0.4219273945731289, 1.344039258992795, 2.312211475443340, 2.608133670226866, 3.852408798052976, 4.661939659377428, 4.966514721687253, 5.536933668953543, 6.010701258569859, 6.665511654217299, 7.409576898559139, 7.817251857634412, 8.499426667409493, 9.093052244157227, 9.755284676858554, 10.46114146116302, 10.85659713247335, 11.18991862828929, 11.66270341462914, 12.41438859839510, 12.83134403418036, 13.23637928801482, 14.11256618731802, 14.35565408538129, 15.12184126788774
