| L(s) = 1 | − 20·7-s + 70·13-s + 56·19-s − 125·25-s − 308·31-s − 110·37-s − 520·43-s + 57·49-s − 182·61-s − 880·67-s + 1.19e3·73-s − 884·79-s − 1.40e3·91-s − 1.33e3·97-s − 1.82e3·103-s + 646·109-s + ⋯ |
| L(s) = 1 | − 1.07·7-s + 1.49·13-s + 0.676·19-s − 25-s − 1.78·31-s − 0.488·37-s − 1.84·43-s + 0.166·49-s − 0.382·61-s − 1.60·67-s + 1.90·73-s − 1.25·79-s − 1.61·91-s − 1.39·97-s − 1.74·103-s + 0.567·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 520 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 182 T + p^{3} T^{2} \) |
| 67 | \( 1 + 880 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 + 884 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
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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
−9.809638470798279571580764215477, −9.100336199545570600055188389828, −8.167603860032875616161123613038, −7.07831348933714242290584814039, −6.22063549463307429132747192529, −5.40351085663299133029438046366, −3.88249231317028251229726666241, −3.19503281087923466256842789439, −1.58047028685932102623093765571, 0,
1.58047028685932102623093765571, 3.19503281087923466256842789439, 3.88249231317028251229726666241, 5.40351085663299133029438046366, 6.22063549463307429132747192529, 7.07831348933714242290584814039, 8.167603860032875616161123613038, 9.100336199545570600055188389828, 9.809638470798279571580764215477
