| L(s) = 1 | − i·2-s − 4.47·3-s + 3·4-s + 4.47i·6-s + 7i·7-s − 7i·8-s + 11.0·9-s + 2·11-s − 13.4·12-s + 13.4·13-s + 7·14-s + 5·16-s + 26.8·17-s − 11.0i·18-s − 13.4i·19-s + ⋯ |
| L(s) = 1 | − 0.5i·2-s − 1.49·3-s + 0.750·4-s + 0.745i·6-s + i·7-s − 0.875i·8-s + 1.22·9-s + 0.181·11-s − 1.11·12-s + 1.03·13-s + 0.5·14-s + 0.312·16-s + 1.57·17-s − 0.611i·18-s − 0.706i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.16414 - 0.274816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16414 - 0.274816i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 2 | \( 1 + iT - 4T^{2} \) |
| 3 | \( 1 + 4.47T + 9T^{2} \) |
| 11 | \( 1 - 2T + 121T^{2} \) |
| 13 | \( 1 - 13.4T + 169T^{2} \) |
| 17 | \( 1 - 26.8T + 289T^{2} \) |
| 19 | \( 1 + 13.4iT - 361T^{2} \) |
| 23 | \( 1 + 26iT - 529T^{2} \) |
| 29 | \( 1 - 22T + 841T^{2} \) |
| 31 | \( 1 - 53.6iT - 961T^{2} \) |
| 37 | \( 1 - 14iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 26.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 34iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 40.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 14iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 62T + 5.04e3T^{2} \) |
| 73 | \( 1 - 53.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38T + 6.24e3T^{2} \) |
| 83 | \( 1 + 40.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−12.24671482558107203002607078368, −11.42884421993973186760037433522, −10.77341009491459984782206842521, −9.820927439780845757599679193370, −8.331276078610471621449477997621, −6.71022549898914785469855203783, −6.08600948646534096598894660767, −4.98182056837784220657325408682, −3.09657233747427090374506067957, −1.20380124664691249088974293876,
1.18156816150138094970852961179, 3.77267025881394423684502810258, 5.44352610278929527852537745025, 6.10124399061851031328314428532, 7.13043378388297136544258876790, 8.021195763622993339379366735371, 9.958287546814530924725198047601, 10.74777298657861627837921488274, 11.52249409513691334471928980687, 12.21817317452212046828055290377
