| L(s) = 1 | + 4·5-s − 7-s − 6·11-s − 2·13-s + 17-s − 4·23-s + 11·25-s − 8·29-s − 4·35-s + 4·37-s + 2·41-s + 8·43-s − 8·47-s + 49-s + 6·53-s − 24·55-s − 4·59-s − 8·61-s − 8·65-s + 16·67-s + 4·71-s + 10·73-s + 6·77-s + 12·79-s + 12·83-s + 4·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 1.80·11-s − 0.554·13-s + 0.242·17-s − 0.834·23-s + 11/5·25-s − 1.48·29-s − 0.676·35-s + 0.657·37-s + 0.312·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 3.23·55-s − 0.520·59-s − 1.02·61-s − 0.992·65-s + 1.95·67-s + 0.474·71-s + 1.17·73-s + 0.683·77-s + 1.35·79-s + 1.31·83-s + 0.433·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.211469686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.211469686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.02321326696637, −15.18844594407115, −14.82220677544848, −13.97119104731323, −13.72700707673853, −13.11798420514030, −12.67478498013853, −12.30487208686938, −11.08414770994151, −10.83525780656352, −10.03879521460483, −9.757191075392459, −9.342452650742230, −8.504824921564160, −7.754520016708101, −7.324041925769158, −6.328533806630583, −5.997051849460608, −5.214183990787652, −5.054764927741063, −3.876273633654207, −2.918489608141673, −2.334452261273633, −1.880688224207089, −0.5973585241864654,
0.5973585241864654, 1.880688224207089, 2.334452261273633, 2.918489608141673, 3.876273633654207, 5.054764927741063, 5.214183990787652, 5.997051849460608, 6.328533806630583, 7.324041925769158, 7.754520016708101, 8.504824921564160, 9.342452650742230, 9.757191075392459, 10.03879521460483, 10.83525780656352, 11.08414770994151, 12.30487208686938, 12.67478498013853, 13.11798420514030, 13.72700707673853, 13.97119104731323, 14.82220677544848, 15.18844594407115, 16.02321326696637
