| L(s) = 1 | − 10·3-s + 73·9-s + 18·11-s + 90·17-s − 106·19-s − 125·25-s − 460·27-s − 180·33-s − 522·41-s + 290·43-s − 343·49-s − 900·51-s + 1.06e3·57-s − 846·59-s + 70·67-s + 430·73-s + 1.25e3·75-s + 2.62e3·81-s + 1.35e3·83-s − 1.02e3·89-s − 1.91e3·97-s + 1.31e3·99-s − 1.71e3·107-s − 270·113-s + ⋯ |
| L(s) = 1 | − 1.92·3-s + 2.70·9-s + 0.493·11-s + 1.28·17-s − 1.27·19-s − 25-s − 3.27·27-s − 0.949·33-s − 1.98·41-s + 1.02·43-s − 49-s − 2.47·51-s + 2.46·57-s − 1.86·59-s + 0.127·67-s + 0.689·73-s + 1.92·75-s + 3.60·81-s + 1.78·83-s − 1.22·89-s − 1.99·97-s + 1.33·99-s − 1.54·107-s − 0.224·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 10 T + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 - 90 T + p^{3} T^{2} \) |
| 19 | \( 1 + 106 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + p^{3} T^{2} \) |
| 41 | \( 1 + 522 T + p^{3} T^{2} \) |
| 43 | \( 1 - 290 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + 846 T + p^{3} T^{2} \) |
| 61 | \( 1 + p^{3} T^{2} \) |
| 67 | \( 1 - 70 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 - 1350 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1910 T + p^{3} T^{2} \) |
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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
−11.15037777711928462517360568670, −10.37931810579069264128929838950, −9.542368770972581575816419132631, −7.912456702358423037405338254607, −6.74204917406960803438714721444, −5.99956328171586295315270547567, −5.04685515117283460756782841614, −3.92952753794774786461886072090, −1.47892469641058410408637517631, 0,
1.47892469641058410408637517631, 3.92952753794774786461886072090, 5.04685515117283460756782841614, 5.99956328171586295315270547567, 6.74204917406960803438714721444, 7.912456702358423037405338254607, 9.542368770972581575816419132631, 10.37931810579069264128929838950, 11.15037777711928462517360568670
