| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 2.00·4-s + 2.23i·5-s − 2.44·6-s − 1.37i·7-s + 2.82·8-s + 2.99·9-s + 3.16i·10-s − 12.6i·11-s − 3.46·12-s − 17.1·13-s − 1.93i·14-s − 3.87i·15-s + 4.00·16-s − 17.1i·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 − 0.195i·7-s + 0.353·8-s + 0.333·9-s + 0.316i·10-s − 1.14i·11-s − 0.288·12-s − 1.32·13-s − 0.138i·14-s − 0.258i·15-s + 0.250·16-s − 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.838471296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.838471296\) |
| \(L(2)\) |
|
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 | \( 1 + (-22.4 + 4.85i)T \) |
| good | 7 | \( 1 + 1.37iT - 49T^{2} \) |
| 11 | \( 1 + 12.6iT - 121T^{2} \) |
| 13 | \( 1 + 17.1T + 169T^{2} \) |
| 17 | \( 1 + 17.1iT - 289T^{2} \) |
| 19 | \( 1 + 5.73iT - 361T^{2} \) |
| 29 | \( 1 - 19.0T + 841T^{2} \) |
| 31 | \( 1 - 27.6T + 961T^{2} \) |
| 37 | \( 1 + 28.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 40.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 36.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 72.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 102. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 83.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 53.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 6.09T + 5.32e3T^{2} \) |
| 79 | \( 1 - 16.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 71.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 47.4iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32214041998222785266011692775, −9.410270325316784559393628417911, −8.163506878241468704354952924496, −7.07770759535780162919553632767, −6.57638697822208245709184040665, −5.37581521469734244875299265302, −4.77849323686033224645890866090, −3.43796219524850813596571454486, −2.46953388653665432135062114825, −0.56193342436362152182539466313,
1.47943384772277361956431096277, 2.77771063494174823838579245773, 4.33983539829354429754332324695, 4.86301510735332940481210362294, 5.82119931058437307363257649622, 6.82912230582467131774338258259, 7.57837327233663524430560047362, 8.709916614058026779810957408950, 9.931017154115471619023241645191, 10.34145472871227226210009032883
