L(s) = 1 | − 4i·2-s + (14.2 + 6.20i)3-s − 16·4-s − 97.9·5-s + (24.8 − 57.1i)6-s − 18.8i·7-s + 64i·8-s + (165. + 177. i)9-s + 391. i·10-s − 79.1·11-s + (−228. − 99.3i)12-s + 704.·13-s − 75.3·14-s + (−1.40e3 − 608. i)15-s + 256·16-s + 2.09e3·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.917 + 0.398i)3-s − 0.5·4-s − 1.75·5-s + (0.281 − 0.648i)6-s − 0.145i·7-s + 0.353i·8-s + (0.682 + 0.730i)9-s + 1.23i·10-s − 0.197·11-s + (−0.458 − 0.199i)12-s + 1.15·13-s − 0.102·14-s + (−1.60 − 0.697i)15-s + 0.250·16-s + 1.76·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.929857629\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.929857629\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 + (-14.2 - 6.20i)T \) |
| 23 | \( 1 + (-392. + 2.50e3i)T \) |
good | 5 | \( 1 + 97.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 18.8iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 79.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 704.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.64e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 7.95e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.44e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.22e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.11e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.01e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.68e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.99e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.83e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.11e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 1.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.22e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5iT - 8.58e9T^{2} \) |
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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
−12.13205209030244922595987552398, −11.07842339441688443628295438480, −10.31835097285829553688164737353, −8.855730990427017442137928530501, −8.216080424011094563659120676094, −7.20118835641970124058932840143, −4.87215792074572090940814837009, −3.76091533793709771640137575993, −3.04771401161699384117774191913, −0.926197864585291640443980126452,
0.946647988551374564193777287912, 3.34233277444636995111210955919, 4.08564446587622860742431411989, 5.93656710223270855586787006832, 7.48479805168911635157028266940, 7.890181519952482207292018235320, 8.724232793094963102919299223362, 10.08094022830453547391547464077, 11.65675869810517038930735008341, 12.39078059730794961847163152976