L(s) = 1 | + (1 + i)2-s + 2i·4-s + 5i·5-s + (2 + 2i)7-s + (−2 + 2i)8-s + (−5 + 5i)10-s + 8·11-s + (3 − 3i)13-s + 4i·14-s − 4·16-s + (−7 − 7i)17-s − 20i·19-s − 10·20-s + (8 + 8i)22-s + (2 − 2i)23-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + 0.5i·4-s + i·5-s + (0.285 + 0.285i)7-s + (−0.250 + 0.250i)8-s + (−0.5 + 0.5i)10-s + 0.727·11-s + (0.230 − 0.230i)13-s + 0.285i·14-s − 0.250·16-s + (−0.411 − 0.411i)17-s − 1.05i·19-s − 0.5·20-s + (0.363 + 0.363i)22-s + (0.0869 − 0.0869i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31917 + 1.04401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31917 + 1.04401i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
good | 7 | \( 1 + (-2 - 2i)T + 49iT^{2} \) |
| 11 | \( 1 - 8T + 121T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 169iT^{2} \) |
| 17 | \( 1 + (7 + 7i)T + 289iT^{2} \) |
| 19 | \( 1 + 20iT - 361T^{2} \) |
| 23 | \( 1 + (-2 + 2i)T - 529iT^{2} \) |
| 29 | \( 1 + 40iT - 841T^{2} \) |
| 31 | \( 1 - 52T + 961T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (42 - 42i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-18 - 18i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (53 - 53i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 20iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-62 - 62i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 28T + 5.04e3T^{2} \) |
| 73 | \( 1 + (47 - 47i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + (18 - 18i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 80iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (63 + 63i)T + 9.40e3iT^{2} \) |
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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
−14.13431962902395914366131936347, −13.35405931161454968575998111711, −11.87787309647466466892041219742, −11.13697289022499561290794207190, −9.683088508157704384785666121113, −8.299684347999981198896119438282, −7.00971697305070295836234339632, −6.07898401272303230111741068109, −4.43971308324919386185934448900, −2.79394477055932692532199011707,
1.48102558081890001302219738510, 3.82730273436598180181313406219, 4.99702136963147333187798867256, 6.43122587510357414693953954050, 8.192394694420734508389404915960, 9.286388328972080110546009079123, 10.51396192154898461426291912661, 11.74529163801427426859045693301, 12.50323212053148142590062758371, 13.57316499754929552636352085392