L(s) = 1 | − 1.41·2-s + 1.73·3-s + 2.00·4-s + 2.23i·5-s − 2.44·6-s + 0.411i·7-s − 2.82·8-s + 2.99·9-s − 3.16i·10-s − 6.98i·11-s + 3.46·12-s − 7.65·13-s − 0.581i·14-s + 3.87i·15-s + 4.00·16-s − 22.0i·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 + 0.0587i·7-s − 0.353·8-s + 0.333·9-s − 0.316i·10-s − 0.635i·11-s + 0.288·12-s − 0.589·13-s − 0.0415i·14-s + 0.258i·15-s + 0.250·16-s − 1.29i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.448090529\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.448090529\) |
\(L(2)\) |
|
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 | \( 1 + (-17.3 + 15.0i)T \) |
good | 7 | \( 1 - 0.411iT - 49T^{2} \) |
| 11 | \( 1 + 6.98iT - 121T^{2} \) |
| 13 | \( 1 + 7.65T + 169T^{2} \) |
| 17 | \( 1 + 22.0iT - 289T^{2} \) |
| 19 | \( 1 + 8.31iT - 361T^{2} \) |
| 29 | \( 1 - 32.6T + 841T^{2} \) |
| 31 | \( 1 + 17.7T + 961T^{2} \) |
| 37 | \( 1 + 22.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 7.58iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 49.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 17.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 39.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 70.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 26.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 62.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 35.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 143. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 73.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 95.9iT - 9.40e3T^{2} \) |
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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
−10.07684116356155498417120685696, −9.141054351007107091305944228738, −8.635732780643170506920027125030, −7.45135875512848624668168877301, −7.01822629536105277231050631339, −5.81375558853240192151985707316, −4.56447247872499075288375693069, −3.10786763013982257052488058609, −2.40496655286714357002021076956, −0.66627820589697966366267889349,
1.25672705765105750023486853204, 2.39387528891924465341535729068, 3.72420796735117833540506618052, 4.84954369496930069096492014827, 6.07586362525930550603050660112, 7.17090933616893153710117208471, 7.892499141199764802632972154688, 8.699744979094080627904223923120, 9.455933734993917024466351386850, 10.18820926788673967262615954060