L(s) = 1 | + 1.41·2-s + (−1.5 − 0.866i)3-s + (1.41 − 1.73i)5-s + (−2.12 − 1.22i)6-s − 2.44i·7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (2.00 − 2.44i)10-s − 4.24·11-s − 3.46i·14-s + (−3.62 + 1.37i)15-s − 4.00·16-s − 1.73i·17-s + (2.12 + 3.67i)18-s − 19-s + ⋯ |
L(s) = 1 | + 1.00·2-s + (−0.866 − 0.499i)3-s + (0.632 − 0.774i)5-s + (−0.866 − 0.499i)6-s − 0.925i·7-s − 0.999·8-s + (0.5 + 0.866i)9-s + (0.632 − 0.774i)10-s − 1.27·11-s − 0.925i·14-s + (−0.935 + 0.354i)15-s − 1.00·16-s − 0.420i·17-s + (0.499 + 0.866i)18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.488687 - 1.12743i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.488687 - 1.12743i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 0.866i)T \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| 31 | \( 1 + (5 - 2.44i)T \) |
good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 1.73iT - 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 8.66iT - 53T^{2} \) |
| 59 | \( 1 + 1.73iT - 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 - 7.34iT - 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 2.44iT - 79T^{2} \) |
| 83 | \( 1 - 1.73iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−10.73663168976210347247846736549, −10.13944129107790266155472606841, −8.876406654021781894161980085962, −7.79774070189195816038988813034, −6.67029574059283600554393070929, −5.74289761465911865867311158741, −4.96264023085114821488600139634, −4.29382971525236152049031128251, −2.49967174654146032635644146557, −0.59608295567762393972429543406,
2.48817030099019428700650332475, 3.61597811530844152662773706597, 4.90143356290354438870512202919, 5.72371630712474767768126830081, 6.07494286464426842858975542391, 7.44405634927717377158534191282, 8.956367551019387164822821163538, 9.727432081334997665540514796738, 10.65466412653943714869029777924, 11.40359890214793686578243849882