L(s) = 1 | + 3·3-s − 4·5-s − 7-s + 6·9-s − 11-s − 12·15-s + 6·19-s − 3·21-s + 7·23-s + 11·25-s + 9·27-s + 4·29-s + 7·31-s − 3·33-s + 4·35-s + 9·37-s + 3·41-s + 4·43-s − 24·45-s + 7·47-s + 49-s + 4·55-s + 18·57-s + 10·59-s − 61-s − 6·63-s − 67-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 1.78·5-s − 0.377·7-s + 2·9-s − 0.301·11-s − 3.09·15-s + 1.37·19-s − 0.654·21-s + 1.45·23-s + 11/5·25-s + 1.73·27-s + 0.742·29-s + 1.25·31-s − 0.522·33-s + 0.676·35-s + 1.47·37-s + 0.468·41-s + 0.609·43-s − 3.57·45-s + 1.02·47-s + 1/7·49-s + 0.539·55-s + 2.38·57-s + 1.30·59-s − 0.128·61-s − 0.755·63-s − 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.484451487\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.484451487\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.14409307566394, −13.60214235519261, −13.03784871063231, −12.75993808909923, −12.01259143310543, −11.73484338386035, −11.02665030551027, −10.56325229560597, −9.788048507756571, −9.455550116819619, −8.819442423754464, −8.434462544992730, −7.925953150439884, −7.552873531542098, −7.129654822119933, −6.653906675969841, −5.667511864035218, −4.770521831396341, −4.433595394601589, −3.761559333686168, −3.341466175528857, −2.754372237299297, −2.505483216299008, −1.091088064812103, −0.7607720054422026,
0.7607720054422026, 1.091088064812103, 2.505483216299008, 2.754372237299297, 3.341466175528857, 3.761559333686168, 4.433595394601589, 4.770521831396341, 5.667511864035218, 6.653906675969841, 7.129654822119933, 7.552873531542098, 7.925953150439884, 8.434462544992730, 8.819442423754464, 9.455550116819619, 9.788048507756571, 10.56325229560597, 11.02665030551027, 11.73484338386035, 12.01259143310543, 12.75993808909923, 13.03784871063231, 13.60214235519261, 14.14409307566394