| 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
−9.902447220276869551716774462002, −9.039563348056528658256834650775, −7.70850919948912657872804950326, −7.07943981898390846863713326420, −5.81199614899744897054865255670, −5.15980766807950767393081042988, −3.85905229202199315357388561496, −2.95200846062194049193019151785, −1.66145096760805090630678750778, 0,
1.66145096760805090630678750778, 2.95200846062194049193019151785, 3.85905229202199315357388561496, 5.15980766807950767393081042988, 5.81199614899744897054865255670, 7.07943981898390846863713326420, 7.70850919948912657872804950326, 9.039563348056528658256834650775, 9.902447220276869551716774462002
