L(s) = 1 | + 1.41·5-s + 4·7-s + 5.65·11-s + 4.24·13-s − 6·17-s − 5.65·19-s + 8·23-s − 2.99·25-s − 4.24·29-s + 4·31-s + 5.65·35-s + 1.41·37-s − 2·41-s + 5.65·43-s − 8·47-s + 9·49-s + 9.89·53-s + 8.00·55-s + 4.24·61-s + 6·65-s − 11.3·67-s + 10·73-s + 22.6·77-s + 12·79-s + 5.65·83-s − 8.48·85-s + 16·89-s + ⋯ |
L(s) = 1 | + 0.632·5-s + 1.51·7-s + 1.70·11-s + 1.17·13-s − 1.45·17-s − 1.29·19-s + 1.66·23-s − 0.599·25-s − 0.787·29-s + 0.718·31-s + 0.956·35-s + 0.232·37-s − 0.312·41-s + 0.862·43-s − 1.16·47-s + 1.28·49-s + 1.35·53-s + 1.07·55-s + 0.543·61-s + 0.744·65-s − 1.38·67-s + 1.17·73-s + 2.57·77-s + 1.35·79-s + 0.620·83-s − 0.920·85-s + 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.554673169\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.554673169\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 9.89T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.83755570287223920399442511480, −6.74472797001150349431123940884, −6.51652409041043753850223347652, −5.72066994805876134102740261380, −4.85095349484092413470050416117, −4.26175440151506504749363871199, −3.66497928552278998574062224136, −2.31949074983506726454665680792, −1.72020307215401050376847078255, −0.997325790249093817029402849287,
0.997325790249093817029402849287, 1.72020307215401050376847078255, 2.31949074983506726454665680792, 3.66497928552278998574062224136, 4.26175440151506504749363871199, 4.85095349484092413470050416117, 5.72066994805876134102740261380, 6.51652409041043753850223347652, 6.74472797001150349431123940884, 7.83755570287223920399442511480