| L(s) = 1 | − 4·2-s + 16·4-s + 47·7-s − 64·8-s − 222·11-s + 101·13-s − 188·14-s + 256·16-s − 162·17-s + 1.68e3·19-s + 888·22-s − 306·23-s − 404·26-s + 752·28-s − 7.89e3·29-s − 8.59e3·31-s − 1.02e3·32-s + 648·34-s + 8.64e3·37-s − 6.74e3·38-s + 1.81e4·41-s + 1.43e4·43-s − 3.55e3·44-s + 1.22e3·46-s + 1.09e3·47-s − 1.45e4·49-s + 1.61e3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.362·7-s − 0.353·8-s − 0.553·11-s + 0.165·13-s − 0.256·14-s + 1/4·16-s − 0.135·17-s + 1.07·19-s + 0.391·22-s − 0.120·23-s − 0.117·26-s + 0.181·28-s − 1.74·29-s − 1.60·31-s − 0.176·32-s + 0.0961·34-s + 1.03·37-s − 0.757·38-s + 1.68·41-s + 1.18·43-s − 0.276·44-s + 0.0852·46-s + 0.0725·47-s − 0.868·49-s + 0.0828·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 47 T + p^{5} T^{2} \) |
| 11 | \( 1 + 222 T + p^{5} T^{2} \) |
| 13 | \( 1 - 101 T + p^{5} T^{2} \) |
| 17 | \( 1 + 162 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1685 T + p^{5} T^{2} \) |
| 23 | \( 1 + 306 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7890 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8593 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8642 T + p^{5} T^{2} \) |
| 41 | \( 1 - 18168 T + p^{5} T^{2} \) |
| 43 | \( 1 - 14351 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1098 T + p^{5} T^{2} \) |
| 53 | \( 1 + 17916 T + p^{5} T^{2} \) |
| 59 | \( 1 + 17610 T + p^{5} T^{2} \) |
| 61 | \( 1 + 21853 T + p^{5} T^{2} \) |
| 67 | \( 1 - 107 T + p^{5} T^{2} \) |
| 71 | \( 1 - 40728 T + p^{5} T^{2} \) |
| 73 | \( 1 - 34706 T + p^{5} T^{2} \) |
| 79 | \( 1 + 69160 T + p^{5} T^{2} \) |
| 83 | \( 1 - 108534 T + p^{5} T^{2} \) |
| 89 | \( 1 + 35040 T + p^{5} T^{2} \) |
| 97 | \( 1 + 823 T + p^{5} T^{2} \) |
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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
−9.603337522561181886952906365244, −9.136217644833093063314705137765, −7.81643574844154170767366092256, −7.47839798532463413952638498360, −6.09802109223475138388468647715, −5.20651882869215390912174839829, −3.80169926138247927141186502581, −2.51261211190531944755256873437, −1.33125305776413623725331592072, 0,
1.33125305776413623725331592072, 2.51261211190531944755256873437, 3.80169926138247927141186502581, 5.20651882869215390912174839829, 6.09802109223475138388468647715, 7.47839798532463413952638498360, 7.81643574844154170767366092256, 9.136217644833093063314705137765, 9.603337522561181886952906365244
