| L(s) = 1 | − 37.4·3-s + 534.·5-s + 1.05e3·7-s − 782.·9-s + 295.·11-s − 5.97e3·13-s − 2.00e4·15-s + 8.60e3·17-s + 6.85e3·19-s − 3.93e4·21-s + 7.10e4·23-s + 2.07e5·25-s + 1.11e5·27-s − 1.34e5·29-s + 1.23e5·31-s − 1.10e4·33-s + 5.62e5·35-s − 1.63e4·37-s + 2.23e5·39-s − 5.59e5·41-s + 8.02e4·43-s − 4.18e5·45-s + 7.86e5·47-s + 2.81e5·49-s − 3.22e5·51-s + 1.46e6·53-s + 1.58e5·55-s + ⋯ |
| L(s) = 1 | − 0.801·3-s + 1.91·5-s + 1.15·7-s − 0.357·9-s + 0.0669·11-s − 0.754·13-s − 1.53·15-s + 0.424·17-s + 0.229·19-s − 0.928·21-s + 1.21·23-s + 2.66·25-s + 1.08·27-s − 1.02·29-s + 0.743·31-s − 0.0536·33-s + 2.21·35-s − 0.0530·37-s + 0.604·39-s − 1.26·41-s + 0.153·43-s − 0.684·45-s + 1.10·47-s + 0.342·49-s − 0.340·51-s + 1.35·53-s + 0.128·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.293446274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293446274\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 + 37.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 534.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.05e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 295.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.97e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 8.60e3T + 4.10e8T^{2} \) |
| 23 | \( 1 - 7.10e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.23e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.63e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.02e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.86e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.46e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.32e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.75e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.33e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.95e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.75e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.02e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.49e7T + 8.07e13T^{2} \) |
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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
−13.13801618247183240716400794641, −11.86135813531384356650235372707, −10.82263268341793170740157367164, −9.859721955204425256833335068437, −8.672366291200635739186976031158, −6.92792172674638159822569332234, −5.58761163664648003385882343442, −5.07397254970131380898691863104, −2.44087952178708572018156984996, −1.13954020283370066529246591284,
1.13954020283370066529246591284, 2.44087952178708572018156984996, 5.07397254970131380898691863104, 5.58761163664648003385882343442, 6.92792172674638159822569332234, 8.672366291200635739186976031158, 9.859721955204425256833335068437, 10.82263268341793170740157367164, 11.86135813531384356650235372707, 13.13801618247183240716400794641
