| L(s) = 1 | + 2·7-s − 3·9-s − 6·11-s − 2·13-s + 6·17-s + 2·19-s + 6·23-s + 6·29-s − 4·31-s − 6·37-s − 2·41-s + 4·43-s − 10·47-s − 3·49-s − 2·53-s − 10·59-s − 10·61-s − 6·63-s − 4·67-s − 16·71-s + 6·73-s − 12·77-s + 9·81-s − 8·83-s + 6·89-s − 4·91-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s − 1.80·11-s − 0.554·13-s + 1.45·17-s + 0.458·19-s + 1.25·23-s + 1.11·29-s − 0.718·31-s − 0.986·37-s − 0.312·41-s + 0.609·43-s − 1.45·47-s − 3/7·49-s − 0.274·53-s − 1.30·59-s − 1.28·61-s − 0.755·63-s − 0.488·67-s − 1.89·71-s + 0.702·73-s − 1.36·77-s + 81-s − 0.878·83-s + 0.635·89-s − 0.419·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.010541165004864335038793834234, −7.86020775957172907109729712996, −6.93423599589363353201596013238, −5.73680595333012578731341415997, −5.22739041241553261209116603001, −4.72626467056231553311741770116, −3.14269786202522870596721822628, −2.82448330488261045122772187462, −1.48396179585067797612111266014, 0,
1.48396179585067797612111266014, 2.82448330488261045122772187462, 3.14269786202522870596721822628, 4.72626467056231553311741770116, 5.22739041241553261209116603001, 5.73680595333012578731341415997, 6.93423599589363353201596013238, 7.86020775957172907109729712996, 8.010541165004864335038793834234
