| L(s) = 1 | + (2.87 + 0.864i)3-s − 9.02·7-s + (7.50 + 4.96i)9-s − 21.8i·11-s − 21.6·13-s − 12.1i·17-s + 3.03·19-s + (−25.9 − 7.80i)21-s − 28.5i·23-s + (17.2 + 20.7i)27-s − 12.0i·29-s + 2.19·31-s + (18.9 − 62.8i)33-s − 0.839·37-s + (−62.2 − 18.7i)39-s + ⋯ |
| L(s) = 1 | + (0.957 + 0.288i)3-s − 1.28·7-s + (0.833 + 0.551i)9-s − 1.98i·11-s − 1.66·13-s − 0.712i·17-s + 0.159·19-s + (−1.23 − 0.371i)21-s − 1.24i·23-s + (0.639 + 0.768i)27-s − 0.415i·29-s + 0.0706·31-s + (0.573 − 1.90i)33-s − 0.0227·37-s + (−1.59 − 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.275990714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.275990714\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.87 - 0.864i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 9.02T + 49T^{2} \) |
| 11 | \( 1 + 21.8iT - 121T^{2} \) |
| 13 | \( 1 + 21.6T + 169T^{2} \) |
| 17 | \( 1 + 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 3.03T + 361T^{2} \) |
| 23 | \( 1 + 28.5iT - 529T^{2} \) |
| 29 | \( 1 + 12.0iT - 841T^{2} \) |
| 31 | \( 1 - 2.19T + 961T^{2} \) |
| 37 | \( 1 + 0.839T + 1.36e3T^{2} \) |
| 41 | \( 1 + 35.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 12.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 9.13iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 80.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 52.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 7.46T + 6.24e3T^{2} \) |
| 83 | \( 1 + 82.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 27.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 114.T + 9.40e3T^{2} \) |
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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
−10.02847190559230067705887892224, −9.306823499686997779584795696388, −8.616817331663581728803617844261, −7.60609164805855289833336768871, −6.70551237938249780059632855213, −5.62159891296621901295222765345, −4.36232497683454956064713823258, −3.15340377045040316727521696149, −2.63407836132214206932112736800, −0.39397244466542847303756604887,
1.82799812680575585094580314195, 2.83134178842794917189147197196, 3.94961114959807808316048676421, 5.03571885971359932780201226067, 6.57974954971090000857719857928, 7.21086701238413447695027067969, 7.88266408140176447832843033408, 9.306493494271516129820434001395, 9.690931226103606904715588686653, 10.22936053078361733901304054632
