| L(s) = 1 | − 12.0i·2-s + (−77.6 − 22.8i)3-s + 109.·4-s + 288. i·5-s + (−276. + 939. i)6-s + 3.02e3·7-s − 4.42e3i·8-s + (5.51e3 + 3.55e3i)9-s + 3.49e3·10-s − 4.41e3i·11-s + (−8.53e3 − 2.51e3i)12-s − 9.09e3·13-s − 3.65e4i·14-s + (6.61e3 − 2.24e4i)15-s − 2.53e4·16-s − 1.07e5i·17-s + ⋯ |
| L(s) = 1 | − 0.755i·2-s + (−0.959 − 0.282i)3-s + 0.429·4-s + 0.462i·5-s + (−0.213 + 0.724i)6-s + 1.25·7-s − 1.07i·8-s + (0.840 + 0.542i)9-s + 0.349·10-s − 0.301i·11-s + (−0.411 − 0.121i)12-s − 0.318·13-s − 0.951i·14-s + (0.130 − 0.443i)15-s − 0.386·16-s − 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.00414 - 1.34259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00414 - 1.34259i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (77.6 + 22.8i)T \) |
| 11 | \( 1 + 4.41e3iT \) |
| good | 2 | \( 1 + 12.0iT - 256T^{2} \) |
| 5 | \( 1 - 288. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 3.02e3T + 5.76e6T^{2} \) |
| 13 | \( 1 + 9.09e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.07e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 5.58e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.45e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.02e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 3.04e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.96e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.62e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.66e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 4.65e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 4.33e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.79e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.96e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.34e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.05e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.67e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 7.34e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 5.71e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.04e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 3.68e7T + 7.83e15T^{2} \) |
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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
−14.52328742715144961304185749860, −12.96377589493071895984011556560, −11.57937069508073193671290651523, −11.24993799482076263190018477892, −10.00623084740196989905951295347, −7.72035409364387686698076753001, −6.42553290875098027562765237318, −4.67696993070946410318674692396, −2.40777177147016736962201504227, −0.860763889873205728859312070386,
1.54949800945950988250106862566, 4.62447827005153945785809271371, 5.72022570114842875222664673871, 7.16802135665009572053434793218, 8.525292486082727707824717844134, 10.51979897039337715479785407791, 11.48516520361539500494720121047, 12.62623487534711513988979026618, 14.59444630467899664523363724814, 15.33204997311483512313827797671
