L(s) = 1 | + (3.10 + 3.10i)2-s + (33.0 − 33.0i)3-s − 236. i·4-s + 205.·6-s + (−974. − 974. i)7-s + (1.53e3 − 1.53e3i)8-s − 2.18e3i·9-s − 4.99e3·11-s + (−7.82e3 − 7.82e3i)12-s + (1.49e4 − 1.49e4i)13-s − 6.05e3i·14-s − 5.10e4·16-s + (−1.17e4 − 1.17e4i)17-s + (6.79e3 − 6.79e3i)18-s − 5.25e4i·19-s + ⋯ |
L(s) = 1 | + (0.194 + 0.194i)2-s + (0.408 − 0.408i)3-s − 0.924i·4-s + 0.158·6-s + (−0.405 − 0.405i)7-s + (0.373 − 0.373i)8-s − 0.333i·9-s − 0.341·11-s + (−0.377 − 0.377i)12-s + (0.523 − 0.523i)13-s − 0.157i·14-s − 0.779·16-s + (−0.141 − 0.141i)17-s + (0.0647 − 0.0647i)18-s − 0.403i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.159299 - 1.36815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159299 - 1.36815i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-33.0 + 33.0i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-3.10 - 3.10i)T + 256iT^{2} \) |
| 7 | \( 1 + (974. + 974. i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + 4.99e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-1.49e4 + 1.49e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (1.17e4 + 1.17e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + 5.25e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (2.45e5 - 2.45e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 - 1.02e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.65e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (2.47e6 + 2.47e6i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 - 3.10e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (1.63e6 - 1.63e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (5.87e6 + 5.87e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (8.41e6 - 8.41e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 + 9.42e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 3.74e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + (1.23e7 + 1.23e7i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 - 1.38e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.14e7 + 2.14e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 4.04e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.85e7 + 1.85e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + 5.33e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-8.55e7 - 8.55e7i)T + 7.83e15iT^{2} \) |
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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
−12.69947541544929665439455320295, −11.11142255244525322274057134250, −10.12215955752775634262404960464, −8.991557997279141860289338611809, −7.52285687931427530605249472698, −6.42017141306664624806507081596, −5.19728648403925012905960279416, −3.51096742752296558464434298306, −1.75667732096177841787111949032, −0.35893166316332525648610055678,
2.18369374363524333037576234560, 3.41088891359944431159373617237, 4.54469130554401911885796265115, 6.31141029290428072529566986250, 7.87997008263782677426157490734, 8.749677471901854757563628420258, 9.991733635731532228802170463643, 11.32052416833107106426724119367, 12.35412455366820141590208419660, 13.32521711466456371324649634336