| L(s) = 1 | + 2·3-s − 5-s + 2·7-s + 9-s − 2·13-s − 2·15-s − 6·17-s + 4·21-s + 6·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 2·35-s − 2·37-s − 4·39-s − 6·41-s − 10·43-s − 45-s − 6·47-s − 3·49-s − 12·51-s + 6·53-s − 12·59-s + 2·61-s + 2·63-s + 2·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 1.52·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 1.68·51-s + 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.251·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.70342498742006188501994585256, −7.06215723111114144957068885167, −6.39347529836665585887623697697, −5.14081916011398824138614106106, −4.74913960233451073492131590460, −3.82183399469630822167941819328, −3.12234796120650330097153328391, −2.31822883421800731675688556606, −1.57301709011122909909496546092, 0,
1.57301709011122909909496546092, 2.31822883421800731675688556606, 3.12234796120650330097153328391, 3.82183399469630822167941819328, 4.74913960233451073492131590460, 5.14081916011398824138614106106, 6.39347529836665585887623697697, 7.06215723111114144957068885167, 7.70342498742006188501994585256
