L(s) = 1 | − 1.41·2-s + 1.73·3-s + 2.00·4-s + 2.23i·5-s − 2.44·6-s − 12.3i·7-s − 2.82·8-s + 2.99·9-s − 3.16i·10-s + 16.3i·11-s + 3.46·12-s − 0.736·13-s + 17.4i·14-s + 3.87i·15-s + 4.00·16-s + 27.6i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.500·4-s + 0.447i·5-s − 0.408·6-s − 1.75i·7-s − 0.353·8-s + 0.333·9-s − 0.316i·10-s + 1.49i·11-s + 0.288·12-s − 0.0566·13-s + 1.24i·14-s + 0.258i·15-s + 0.250·16-s + 1.62i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.585977053\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585977053\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (11.5 + 19.8i)T \) |
good | 7 | \( 1 + 12.3iT - 49T^{2} \) |
| 11 | \( 1 - 16.3iT - 121T^{2} \) |
| 13 | \( 1 + 0.736T + 169T^{2} \) |
| 17 | \( 1 - 27.6iT - 289T^{2} \) |
| 19 | \( 1 - 16.9iT - 361T^{2} \) |
| 29 | \( 1 - 45.7T + 841T^{2} \) |
| 31 | \( 1 - 55.8T + 961T^{2} \) |
| 37 | \( 1 + 4.22iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 23.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 8.31iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 19.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 28.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 61.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 56.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 63.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 22.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 93.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 29.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 151. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 89.5iT - 9.40e3T^{2} \) |
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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
−10.26332658596951619291137728476, −9.821964048962017347107638057403, −8.366087544116413364544530657856, −7.86204650982616014150792866226, −6.96804280499379622383791989307, −6.37475130957463309430344241227, −4.46709120141916845970090975681, −3.81840289227324038219513477023, −2.35939090815727214668928616857, −1.14885499589915474953361038769,
0.792051012330145022921473584883, 2.47607157079192695093547044128, 3.05375646416629006726767454261, 4.85489230302420269400422786519, 5.77215570274617556736963008053, 6.70156346040947193456492327897, 8.054942334785008425129215173303, 8.513005228649156858527417845301, 9.231703453533000728041704306370, 9.768716727200324425465341128024