| L(s) = 1 | − 3-s + 2·5-s + 7-s − 2·9-s − 4·11-s + 13-s − 2·15-s − 5·17-s − 19-s − 21-s + 9·23-s − 25-s + 5·27-s + 5·29-s − 2·31-s + 4·33-s + 2·35-s + 10·37-s − 39-s + 12·41-s − 10·43-s − 4·45-s − 12·47-s − 6·49-s + 5·51-s + 9·53-s − 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s − 1.20·11-s + 0.277·13-s − 0.516·15-s − 1.21·17-s − 0.229·19-s − 0.218·21-s + 1.87·23-s − 1/5·25-s + 0.962·27-s + 0.928·29-s − 0.359·31-s + 0.696·33-s + 0.338·35-s + 1.64·37-s − 0.160·39-s + 1.87·41-s − 1.52·43-s − 0.596·45-s − 1.75·47-s − 6/7·49-s + 0.700·51-s + 1.23·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.969957024073968534263365138808, −7.08284170502121509016570453886, −6.24245502240187913411510784582, −5.82159008891349783075847870968, −4.94434616110895581911363827628, −4.56228649918768272678284423362, −3.01102310628209630355501543949, −2.49707499281623976581581809291, −1.34715030765928874578193019224, 0,
1.34715030765928874578193019224, 2.49707499281623976581581809291, 3.01102310628209630355501543949, 4.56228649918768272678284423362, 4.94434616110895581911363827628, 5.82159008891349783075847870968, 6.24245502240187913411510784582, 7.08284170502121509016570453886, 7.969957024073968534263365138808
