| L(s) = 1 | + 2·3-s − 2·7-s + 9-s + 4·11-s + 4·19-s − 4·21-s − 2·23-s − 4·27-s + 2·29-s + 8·33-s − 4·37-s − 2·41-s − 6·43-s + 6·47-s − 3·49-s − 4·53-s + 8·57-s + 12·59-s − 10·61-s − 2·63-s − 14·67-s − 4·69-s − 8·71-s − 8·73-s − 8·77-s + 16·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.917·19-s − 0.872·21-s − 0.417·23-s − 0.769·27-s + 0.371·29-s + 1.39·33-s − 0.657·37-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.549·53-s + 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s − 1.71·67-s − 0.481·69-s − 0.949·71-s − 0.936·73-s − 0.911·77-s + 1.80·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.14560823818759, −14.69941908961482, −14.11987577562670, −13.77340512293709, −13.36880471074327, −12.68821156609190, −12.04767700748064, −11.72266286130761, −11.05564706388256, −10.13317084498282, −9.911415044310947, −9.169738700004178, −8.927951999903222, −8.360759452451025, −7.641904107429019, −7.186098075329945, −6.493410064114645, −6.018722908639490, −5.251742978316106, −4.415416913841411, −3.756262743752821, −3.278646782535452, −2.783275476239225, −1.888151047774056, −1.215167276624927, 0,
1.215167276624927, 1.888151047774056, 2.783275476239225, 3.278646782535452, 3.756262743752821, 4.415416913841411, 5.251742978316106, 6.018722908639490, 6.493410064114645, 7.186098075329945, 7.641904107429019, 8.360759452451025, 8.927951999903222, 9.169738700004178, 9.911415044310947, 10.13317084498282, 11.05564706388256, 11.72266286130761, 12.04767700748064, 12.68821156609190, 13.36880471074327, 13.77340512293709, 14.11987577562670, 14.69941908961482, 15.14560823818759
