| L(s) = 1 | + 5-s − 11-s − 5·17-s + 5·19-s + 23-s + 25-s − 5·29-s − 10·31-s + 9·37-s + 6·41-s + 7·43-s + 13·47-s − 2·53-s − 55-s − 3·59-s + 8·61-s + 2·67-s + 9·71-s − 8·73-s + 12·79-s − 10·83-s − 5·85-s + 14·89-s + 5·95-s + 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s − 1.21·17-s + 1.14·19-s + 0.208·23-s + 1/5·25-s − 0.928·29-s − 1.79·31-s + 1.47·37-s + 0.937·41-s + 1.06·43-s + 1.89·47-s − 0.274·53-s − 0.134·55-s − 0.390·59-s + 1.02·61-s + 0.244·67-s + 1.06·71-s − 0.936·73-s + 1.35·79-s − 1.09·83-s − 0.542·85-s + 1.48·89-s + 0.512·95-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-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
| \(L(1)\) |
\(\approx\) |
\(3.078172315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.078172315\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.55824282306839, −12.05679521774950, −11.36954950214738, −11.07163956463413, −10.84257167597877, −10.20103830128908, −9.621476755374902, −9.279531486405559, −8.994358615948859, −8.482139292804607, −7.668639672440444, −7.444012027872709, −7.139044244158074, −6.260143821275354, −6.071499639090727, −5.387886086067140, −5.174859320799032, −4.393173420626137, −4.005558382994538, −3.440066280501499, −2.759682825432467, −2.264767893682432, −1.876116411463873, −0.9826887817769047, −0.5084398219420246,
0.5084398219420246, 0.9826887817769047, 1.876116411463873, 2.264767893682432, 2.759682825432467, 3.440066280501499, 4.005558382994538, 4.393173420626137, 5.174859320799032, 5.387886086067140, 6.071499639090727, 6.260143821275354, 7.139044244158074, 7.444012027872709, 7.668639672440444, 8.482139292804607, 8.994358615948859, 9.279531486405559, 9.621476755374902, 10.20103830128908, 10.84257167597877, 11.07163956463413, 11.36954950214738, 12.05679521774950, 12.55824282306839
