| L(s) = 1 | − 0.112i·2-s + 1.73·3-s + 3.98·4-s − 0.195i·6-s + (1.98 − 6.71i)7-s − 0.902i·8-s + 2.99·9-s − 15.8·11-s + 6.90·12-s + 13.3·13-s + (−0.758 − 0.224i)14-s + 15.8·16-s + 15.6·17-s − 0.338i·18-s − 30.8i·19-s + ⋯ |
| L(s) = 1 | − 0.0564i·2-s + 0.577·3-s + 0.996·4-s − 0.0326i·6-s + (0.283 − 0.959i)7-s − 0.112i·8-s + 0.333·9-s − 1.44·11-s + 0.575·12-s + 1.02·13-s + (−0.0541 − 0.0160i)14-s + 0.990·16-s + 0.922·17-s − 0.0188i·18-s − 1.62i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.789683340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.789683340\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.98 + 6.71i)T \) |
| good | 2 | \( 1 + 0.112iT - 4T^{2} \) |
| 11 | \( 1 + 15.8T + 121T^{2} \) |
| 13 | \( 1 - 13.3T + 169T^{2} \) |
| 17 | \( 1 - 15.6T + 289T^{2} \) |
| 19 | \( 1 + 30.8iT - 361T^{2} \) |
| 23 | \( 1 + 3.63iT - 529T^{2} \) |
| 29 | \( 1 + 14.5T + 841T^{2} \) |
| 31 | \( 1 - 11.3iT - 961T^{2} \) |
| 37 | \( 1 - 17.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 27.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 12.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 80.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 55.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 79.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 94.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 103. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 113.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 20.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 1.27T + 6.24e3T^{2} \) |
| 83 | \( 1 - 19.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 12.4T + 9.40e3T^{2} \) |
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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
−10.72983511200048725293662105900, −9.858884376759259435863964695899, −8.595745791742966567474267559674, −7.68585865102460752471138272669, −7.18774684986460882145296578942, −6.02140007040064862364795061124, −4.80812432989948422717650130455, −3.46108836865074802819523881438, −2.55117490243491324420017229029, −1.10157326504762136880241124765,
1.67271892684871877031842265297, 2.67471804426838225758189665443, 3.70095163969323636775552719770, 5.51359442991359920574869429520, 5.91954514403666863002104423257, 7.41213404591706639118698085874, 7.993517497824288096126434390303, 8.774480372103362557093549619816, 10.05391960736580706408186126846, 10.65427145569338230945603577444
