L(s) = 1 | + (−0.450 + 1.34i)2-s + i·3-s + (−1.59 − 1.20i)4-s + (−1.34 − 0.450i)6-s + 2.64·7-s + (2.33 − 1.59i)8-s − 9-s − 1.51i·11-s + (1.20 − 1.59i)12-s + 3.87i·13-s + (−1.18 + 3.54i)14-s + (1.08 + 3.84i)16-s + 3.31·17-s + (0.450 − 1.34i)18-s + 7.08i·19-s + ⋯ |
L(s) = 1 | + (−0.318 + 0.947i)2-s + 0.577i·3-s + (−0.797 − 0.603i)4-s + (−0.547 − 0.183i)6-s + 0.998·7-s + (0.825 − 0.563i)8-s − 0.333·9-s − 0.456i·11-s + (0.348 − 0.460i)12-s + 1.07i·13-s + (−0.317 + 0.946i)14-s + (0.271 + 0.962i)16-s + 0.803·17-s + (0.106 − 0.315i)18-s + 1.62i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.580760 + 1.09968i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.580760 + 1.09968i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.450 - 1.34i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 + 1.51iT - 11T^{2} \) |
| 13 | \( 1 - 3.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 - 7.08iT - 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.18iT - 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 - 7.87iT - 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 - 1.01iT - 43T^{2} \) |
| 47 | \( 1 + 7.08T + 47T^{2} \) |
| 53 | \( 1 - 4.50iT - 53T^{2} \) |
| 59 | \( 1 + 6.79iT - 59T^{2} \) |
| 61 | \( 1 + 3.60iT - 61T^{2} \) |
| 67 | \( 1 - 1.01iT - 67T^{2} \) |
| 71 | \( 1 + 6.72T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 7.36T + 79T^{2} \) |
| 83 | \( 1 - 7.74iT - 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−10.83753184063489894435602937065, −9.876962666735909855418351663180, −9.145416366014996883600986693796, −8.239397035737924239656284424862, −7.62737890401704711843604772332, −6.41392736614325215224995099872, −5.49325729170002586813723452417, −4.66439313985363198200895337044, −3.64297498736320779985968351396, −1.51681058281702573856635576134,
0.888008447718660803343565561406, 2.18850875419718800536783374044, 3.29917975632862668681258457054, 4.73064519309181136270615358471, 5.48250708016272044271730996906, 7.21048030403424997239023831354, 7.74985373691166743122817674619, 8.724597621474095274475171309209, 9.454964880441780168858609357331, 10.69203142614528882343799538061