L(s) = 1 | + 3.25·2-s − 21.3·4-s + 25·5-s − 3.69·7-s − 173.·8-s + 81.4·10-s − 121·11-s − 806.·13-s − 12.0·14-s + 118.·16-s − 989.·17-s + 116.·19-s − 534.·20-s − 394.·22-s + 4.45e3·23-s + 625·25-s − 2.62e3·26-s + 78.9·28-s + 109.·29-s + 4.44e3·31-s + 5.94e3·32-s − 3.22e3·34-s − 92.2·35-s − 9.61e3·37-s + 378.·38-s − 4.34e3·40-s + 2.76e3·41-s + ⋯ |
L(s) = 1 | + 0.575·2-s − 0.668·4-s + 0.447·5-s − 0.0284·7-s − 0.960·8-s + 0.257·10-s − 0.301·11-s − 1.32·13-s − 0.0163·14-s + 0.115·16-s − 0.830·17-s + 0.0737·19-s − 0.299·20-s − 0.173·22-s + 1.75·23-s + 0.200·25-s − 0.762·26-s + 0.0190·28-s + 0.0242·29-s + 0.830·31-s + 1.02·32-s − 0.478·34-s − 0.0127·35-s − 1.15·37-s + 0.0424·38-s − 0.429·40-s + 0.256·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.858351465\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.858351465\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 3.25T + 32T^{2} \) |
| 7 | \( 1 + 3.69T + 1.68e4T^{2} \) |
| 13 | \( 1 + 806.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 989.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 116.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 109.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.61e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.76e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.79e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e5T + 8.58e9T^{2} \) |
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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
−10.03219969439000927382678977282, −9.293468875962624734204141048325, −8.527510985921967186869224823130, −7.29753021014668854644081903496, −6.33436206103154832136991900540, −5.12030197926252685361875075478, −4.72793891379736882166101605007, −3.33242556610091646042756115082, −2.31337844419830219374564975330, −0.61749337474304488062183030948,
0.61749337474304488062183030948, 2.31337844419830219374564975330, 3.33242556610091646042756115082, 4.72793891379736882166101605007, 5.12030197926252685361875075478, 6.33436206103154832136991900540, 7.29753021014668854644081903496, 8.527510985921967186869224823130, 9.293468875962624734204141048325, 10.03219969439000927382678977282