| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 2.00·4-s − 2.23i·5-s − 2.44·6-s − 7.36i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s − 16.6i·11-s + 3.46·12-s + 9.09·13-s + 10.4i·14-s − 3.87i·15-s + 4.00·16-s + 4.72i·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 − 1.05i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s − 1.51i·11-s + 0.288·12-s + 0.699·13-s + 0.743i·14-s − 0.258i·15-s + 0.250·16-s + 0.278i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.258979343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258979343\) |
| \(L(2)\) |
|
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 | \( 1 + (18.8 + 13.2i)T \) |
| good | 7 | \( 1 + 7.36iT - 49T^{2} \) |
| 11 | \( 1 + 16.6iT - 121T^{2} \) |
| 13 | \( 1 - 9.09T + 169T^{2} \) |
| 17 | \( 1 - 4.72iT - 289T^{2} \) |
| 19 | \( 1 - 11.5iT - 361T^{2} \) |
| 29 | \( 1 + 8.88T + 841T^{2} \) |
| 31 | \( 1 + 45.3T + 961T^{2} \) |
| 37 | \( 1 + 9.59iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 20.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 21.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 70.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 47.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 16.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 11.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 2.91T + 5.04e3T^{2} \) |
| 73 | \( 1 - 136.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 37.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 10.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 70.0iT - 9.40e3T^{2} \) |
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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
−9.898364429408040627714376802507, −8.974736334471965036363503417686, −8.273620969615152846676333945322, −7.69033016589073313420036219730, −6.54188826916912929481248578859, −5.64389527629674183685332873898, −4.06316073030171196183122483472, −3.33751534114634300365984406911, −1.70415441404021692965456273466, −0.51722042831220965729733791547,
1.76618191576823890070815516748, 2.58231901033167493648549137548, 3.84129649866501915204217504491, 5.23218100609105943067256799938, 6.34677648518434994936554482676, 7.28432977516661244056813292265, 7.968613934588141824161820129250, 9.099161714693207260084535848243, 9.417650036251092831047833138684, 10.39390953310531373505968157060
