| L(s) = 1 | + 5-s − 11-s + 6·13-s − 5·17-s − 7·19-s + 5·23-s + 25-s + 3·29-s + 6·31-s − 5·37-s − 6·41-s + 7·43-s + 11·47-s + 8·53-s − 55-s + 9·59-s + 2·61-s + 6·65-s + 8·67-s − 5·71-s + 16·73-s + 8·79-s + 6·83-s − 5·85-s − 14·89-s − 7·95-s − 3·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 1.66·13-s − 1.21·17-s − 1.60·19-s + 1.04·23-s + 1/5·25-s + 0.557·29-s + 1.07·31-s − 0.821·37-s − 0.937·41-s + 1.06·43-s + 1.60·47-s + 1.09·53-s − 0.134·55-s + 1.17·59-s + 0.256·61-s + 0.744·65-s + 0.977·67-s − 0.593·71-s + 1.87·73-s + 0.900·79-s + 0.658·83-s − 0.542·85-s − 1.48·89-s − 0.718·95-s − 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.562000592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.562000592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.55069962815813, −12.05923400621674, −11.44128887823117, −10.95810583059664, −10.68817124025859, −10.42483190468378, −9.756857767264613, −9.174731066023553, −8.729638215365476, −8.452976568960823, −8.186162919382845, −7.266789896034260, −6.822060202216536, −6.442899245184050, −6.143458079081725, −5.400159769304526, −5.120797537358688, −4.285222309103921, −4.084672715184681, −3.488037129085368, −2.697582700778273, −2.352610906764181, −1.784035280120533, −1.007480675782267, −0.5468843473576596,
0.5468843473576596, 1.007480675782267, 1.784035280120533, 2.352610906764181, 2.697582700778273, 3.488037129085368, 4.084672715184681, 4.285222309103921, 5.120797537358688, 5.400159769304526, 6.143458079081725, 6.442899245184050, 6.822060202216536, 7.266789896034260, 8.186162919382845, 8.452976568960823, 8.729638215365476, 9.174731066023553, 9.756857767264613, 10.42483190468378, 10.68817124025859, 10.95810583059664, 11.44128887823117, 12.05923400621674, 12.55069962815813
