| L(s) = 1 | − 8.48·3-s − 9.89·5-s − 171·9-s − 308·11-s − 912.·13-s + 83.9·15-s − 451.·17-s − 2.61e3·19-s + 2.32e3·23-s − 3.02e3·25-s + 3.51e3·27-s − 6.48e3·29-s + 814.·31-s + 2.61e3·33-s − 1.22e4·37-s + 7.73e3·39-s − 527.·41-s + 1.50e4·43-s + 1.69e3·45-s + 1.60e3·47-s + 3.82e3·51-s + 1.07e4·53-s + 3.04e3·55-s + 2.22e4·57-s − 5.21e4·59-s − 3.67e4·61-s + 9.03e3·65-s + ⋯ |
| L(s) = 1 | − 0.544·3-s − 0.177·5-s − 0.703·9-s − 0.767·11-s − 1.49·13-s + 0.0963·15-s − 0.378·17-s − 1.66·19-s + 0.916·23-s − 0.968·25-s + 0.927·27-s − 1.43·29-s + 0.152·31-s + 0.417·33-s − 1.46·37-s + 0.814·39-s − 0.0490·41-s + 1.23·43-s + 0.124·45-s + 0.105·47-s + 0.206·51-s + 0.527·53-s + 0.135·55-s + 0.905·57-s − 1.94·59-s − 1.26·61-s + 0.265·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1351680535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1351680535\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 8.48T + 243T^{2} \) |
| 5 | \( 1 + 9.89T + 3.12e3T^{2} \) |
| 11 | \( 1 + 308T + 1.61e5T^{2} \) |
| 13 | \( 1 + 912.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 451.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 814.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 527.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.07e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.68e4T + 8.58e9T^{2} \) |
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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
−9.538500728963096675851938040507, −8.704365487173789386947342731751, −7.75661846146683122212832197168, −6.96533269203133164127658534458, −5.91424318815298018286745281914, −5.15229533904992613127727957297, −4.27310220999585426020251535955, −2.89582777361489780304397027971, −1.99728271792033368857777113143, −0.16078664080807025861658673382,
0.16078664080807025861658673382, 1.99728271792033368857777113143, 2.89582777361489780304397027971, 4.27310220999585426020251535955, 5.15229533904992613127727957297, 5.91424318815298018286745281914, 6.96533269203133164127658534458, 7.75661846146683122212832197168, 8.704365487173789386947342731751, 9.538500728963096675851938040507
