| L(s) = 1 | + 508·7-s − 1.46e4·13-s + 5.74e4·19-s − 7.81e4·25-s − 1.78e5·31-s + 2.79e5·37-s − 1.03e6·43-s − 5.65e5·49-s − 3.53e6·61-s + 3.85e5·67-s − 6.27e6·73-s − 8.76e6·79-s − 7.42e6·91-s + 1.22e7·97-s − 8.02e6·103-s − 1.68e7·109-s + ⋯ |
| L(s) = 1 | + 0.559·7-s − 1.84·13-s + 1.92·19-s − 25-s − 1.07·31-s + 0.907·37-s − 1.98·43-s − 0.686·49-s − 1.99·61-s + 0.156·67-s − 1.88·73-s − 1.99·79-s − 1.03·91-s + 1.36·97-s − 0.723·103-s − 1.24·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{7} T^{2} \) |
| 7 | \( 1 - 508 T + p^{7} T^{2} \) |
| 11 | \( 1 + p^{7} T^{2} \) |
| 13 | \( 1 + 14614 T + p^{7} T^{2} \) |
| 17 | \( 1 + p^{7} T^{2} \) |
| 19 | \( 1 - 57448 T + p^{7} T^{2} \) |
| 23 | \( 1 + p^{7} T^{2} \) |
| 29 | \( 1 + p^{7} T^{2} \) |
| 31 | \( 1 + 178916 T + p^{7} T^{2} \) |
| 37 | \( 1 - 279710 T + p^{7} T^{2} \) |
| 41 | \( 1 + p^{7} T^{2} \) |
| 43 | \( 1 + 1035224 T + p^{7} T^{2} \) |
| 47 | \( 1 + p^{7} T^{2} \) |
| 53 | \( 1 + p^{7} T^{2} \) |
| 59 | \( 1 + p^{7} T^{2} \) |
| 61 | \( 1 + 3535546 T + p^{7} T^{2} \) |
| 67 | \( 1 - 385072 T + p^{7} T^{2} \) |
| 71 | \( 1 + p^{7} T^{2} \) |
| 73 | \( 1 + 6274810 T + p^{7} T^{2} \) |
| 79 | \( 1 + 8763044 T + p^{7} T^{2} \) |
| 83 | \( 1 + p^{7} T^{2} \) |
| 89 | \( 1 + p^{7} T^{2} \) |
| 97 | \( 1 - 12245198 T + p^{7} T^{2} \) |
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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
−11.47479200820600571993827618708, −10.07616381786588444256374234005, −9.358948223154158007414600089657, −7.88115799983055991957965346823, −7.19616643248430871132582093287, −5.57314110895814532960910056828, −4.64348051727966367649628377273, −3.05107249995230713067253980698, −1.65403634759360067525548002393, 0,
1.65403634759360067525548002393, 3.05107249995230713067253980698, 4.64348051727966367649628377273, 5.57314110895814532960910056828, 7.19616643248430871132582093287, 7.88115799983055991957965346823, 9.358948223154158007414600089657, 10.07616381786588444256374234005, 11.47479200820600571993827618708
