L(s) = 1 | + 133.·7-s + 345.·11-s − 405.·13-s + 2.13e3·17-s − 1.31e3·19-s + 808.·23-s − 984.·29-s + 8.01e3·31-s + 9.34e3·37-s − 1.91e4·41-s + 5.83e3·43-s + 8.89e3·47-s + 920.·49-s + 3.05e4·53-s − 4.90e4·59-s − 423.·61-s + 1.04e4·67-s + 7.81e3·71-s − 2.71e4·73-s + 4.60e4·77-s + 3.15e4·79-s + 8.75e3·83-s + 2.15e4·89-s − 5.39e4·91-s − 5.26e4·97-s − 7.28e4·101-s + 2.86e3·103-s + ⋯ |
L(s) = 1 | + 1.02·7-s + 0.861·11-s − 0.664·13-s + 1.78·17-s − 0.835·19-s + 0.318·23-s − 0.217·29-s + 1.49·31-s + 1.12·37-s − 1.77·41-s + 0.481·43-s + 0.587·47-s + 0.0547·49-s + 1.49·53-s − 1.83·59-s − 0.0145·61-s + 0.284·67-s + 0.184·71-s − 0.596·73-s + 0.884·77-s + 0.569·79-s + 0.139·83-s + 0.288·89-s − 0.682·91-s − 0.568·97-s − 0.711·101-s + 0.0265·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.977933841\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.977933841\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 133.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 345.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 405.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 808.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 984.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.34e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.91e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 423.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.81e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.75e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.26e4T + 8.58e9T^{2} \) |
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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
−9.412306862755829076504562939737, −8.396035552642305373496387610299, −7.79953548364807440087310088854, −6.86407820183503947915249388670, −5.85540346152618810663474225536, −4.92817251129582366108317100660, −4.11325681819891330154035832542, −2.93187651098708002583083257459, −1.72108598203957642866515154245, −0.816249180844321590697400541328,
0.816249180844321590697400541328, 1.72108598203957642866515154245, 2.93187651098708002583083257459, 4.11325681819891330154035832542, 4.92817251129582366108317100660, 5.85540346152618810663474225536, 6.86407820183503947915249388670, 7.79953548364807440087310088854, 8.396035552642305373496387610299, 9.412306862755829076504562939737