L(s) = 1 | + 8.71·2-s + 44.0·4-s + 99.6·5-s + 104.·8-s + 868.·10-s + 374.·11-s + 868.·13-s − 495.·16-s − 1.09e3·17-s − 868.·19-s + 4.38e3·20-s + 3.26e3·22-s + 2.97e3·23-s + 6.81e3·25-s + 7.57e3·26-s + 4.51e3·29-s − 9.55e3·31-s − 7.67e3·32-s − 9.55e3·34-s − 5.46e3·37-s − 7.57e3·38-s + 1.04e4·40-s + 9.86e3·41-s + 1.25e4·43-s + 1.64e4·44-s + 2.59e4·46-s − 9.96e3·47-s + ⋯ |
L(s) = 1 | + 1.54·2-s + 1.37·4-s + 1.78·5-s + 0.577·8-s + 2.74·10-s + 0.934·11-s + 1.42·13-s − 0.484·16-s − 0.920·17-s − 0.552·19-s + 2.45·20-s + 1.43·22-s + 1.17·23-s + 2.17·25-s + 2.19·26-s + 0.997·29-s − 1.78·31-s − 1.32·32-s − 1.41·34-s − 0.656·37-s − 0.851·38-s + 1.03·40-s + 0.916·41-s + 1.03·43-s + 1.28·44-s + 1.80·46-s − 0.658·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(8.032512331\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.032512331\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 8.71T + 32T^{2} \) |
| 5 | \( 1 - 99.6T + 3.12e3T^{2} \) |
| 11 | \( 1 - 374.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 868.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 868.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.97e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.96e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.86e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.08e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.33e5T + 8.58e9T^{2} \) |
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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
−10.70082444135937252094094836590, −9.260190183982918571532015724875, −8.854998182208820853177482868653, −6.82587963752568901565701215656, −6.29512335163101504674777760737, −5.60087651655389706781391121257, −4.60144330627534084647586392951, −3.50545723784076952286868138194, −2.33924826763764349662170238856, −1.35622425856082421627766480521,
1.35622425856082421627766480521, 2.33924826763764349662170238856, 3.50545723784076952286868138194, 4.60144330627534084647586392951, 5.60087651655389706781391121257, 6.29512335163101504674777760737, 6.82587963752568901565701215656, 8.854998182208820853177482868653, 9.260190183982918571532015724875, 10.70082444135937252094094836590