| L(s) = 1 | − 2·3-s − 3·7-s + 9-s + 3·11-s − 5·17-s − 2·19-s + 6·21-s + 6·23-s + 4·27-s + 29-s − 31-s − 6·33-s − 37-s − 5·41-s − 43-s − 4·47-s + 2·49-s + 10·51-s + 9·53-s + 4·57-s + 8·59-s + 5·61-s − 3·63-s − 12·69-s − 2·71-s + 4·73-s − 9·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.21·17-s − 0.458·19-s + 1.30·21-s + 1.25·23-s + 0.769·27-s + 0.185·29-s − 0.179·31-s − 1.04·33-s − 0.164·37-s − 0.780·41-s − 0.152·43-s − 0.583·47-s + 2/7·49-s + 1.40·51-s + 1.23·53-s + 0.529·57-s + 1.04·59-s + 0.640·61-s − 0.377·63-s − 1.44·69-s − 0.237·71-s + 0.468·73-s − 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−7.15185086852814066845555660882, −6.58965492643969974539921763368, −6.42047797155490355370625517260, −5.48541868820233631840825678333, −4.84339179025146195346839099774, −4.00421399818620286073507484443, −3.23185038642651944333212159560, −2.23580823591016035403520503836, −0.962944518567001489857217104945, 0,
0.962944518567001489857217104945, 2.23580823591016035403520503836, 3.23185038642651944333212159560, 4.00421399818620286073507484443, 4.84339179025146195346839099774, 5.48541868820233631840825678333, 6.42047797155490355370625517260, 6.58965492643969974539921763368, 7.15185086852814066845555660882
