L(s) = 1 | + (33.6 − 30.2i)2-s − 243·3-s + (218. − 2.03e3i)4-s − 2.34e3i·5-s + (−8.18e3 + 7.34e3i)6-s + (2.28e4 − 3.81e4i)7-s + (−5.42e4 − 7.51e4i)8-s + 5.90e4·9-s + (−7.09e4 − 7.90e4i)10-s + 8.78e5i·11-s + (−5.31e4 + 4.94e5i)12-s − 2.43e6i·13-s + (−3.87e5 − 1.97e6i)14-s + 5.70e5i·15-s + (−4.09e6 − 8.90e5i)16-s − 7.78e6i·17-s + ⋯ |
L(s) = 1 | + (0.743 − 0.668i)2-s − 0.577·3-s + (0.106 − 0.994i)4-s − 0.335i·5-s + (−0.429 + 0.385i)6-s + (0.512 − 0.858i)7-s + (−0.585 − 0.810i)8-s + 0.333·9-s + (−0.224 − 0.249i)10-s + 1.64i·11-s + (−0.0616 + 0.574i)12-s − 1.82i·13-s + (−0.192 − 0.981i)14-s + 0.193i·15-s + (−0.977 − 0.212i)16-s − 1.33i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.542309 + 1.62196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.542309 + 1.62196i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-33.6 + 30.2i)T \) |
| 3 | \( 1 + 243T \) |
| 7 | \( 1 + (-2.28e4 + 3.81e4i)T \) |
good | 5 | \( 1 + 2.34e3iT - 4.88e7T^{2} \) |
| 11 | \( 1 - 8.78e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 + 2.43e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 7.78e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 + 5.19e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.60e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 2.93e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.06e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.32e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 4.43e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 - 9.39e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 3.08e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.68e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.41e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 3.61e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 - 1.64e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 9.96e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 1.66e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 1.88e10iT - 7.47e20T^{2} \) |
| 83 | \( 1 - 8.20e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.00e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 7.38e8iT - 7.15e21T^{2} \) |
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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
−11.42854007980322539175714658428, −10.45259134813000296394814395000, −9.710045538052702167060966216740, −7.77206634519825338239347464461, −6.56204794342993041525337720057, −4.97937384411649791119963177988, −4.56292988671581803574319050444, −2.84938178323366121089403167195, −1.33362383023603026069907268651, −0.35059843634858399574263484464,
1.83370276991294248587791581217, 3.41370413273938973873478805258, 4.69171292536197674723020185195, 5.94060686657596766328445603428, 6.55073515903501914140768329990, 8.121997425503577353930812761359, 9.033919770393471091347043459569, 11.06540081011096804932751109077, 11.57699395849882136537762287540, 12.70221171879919384991759807508