| L(s) = 1 | − 2·3-s + 2·5-s + 7-s + 9-s + 11-s − 4·13-s − 4·15-s − 2·21-s + 8·23-s − 25-s + 4·27-s − 6·29-s − 2·31-s − 2·33-s + 2·35-s − 10·37-s + 8·39-s + 4·43-s + 2·45-s + 2·47-s + 49-s − 10·53-s + 2·55-s − 10·59-s + 8·61-s + 63-s − 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 1.03·15-s − 0.436·21-s + 1.66·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.359·31-s − 0.348·33-s + 0.338·35-s − 1.64·37-s + 1.28·39-s + 0.609·43-s + 0.298·45-s + 0.291·47-s + 1/7·49-s − 1.37·53-s + 0.269·55-s − 1.30·59-s + 1.02·61-s + 0.125·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.68158474603930669465349935865, −7.03249359501250263096101255518, −6.38664649252791786925743548227, −5.53623218973341603191591242782, −5.21581303955537840355157129629, −4.48080927891411339649043282240, −3.26702957402052559665502362400, −2.22483963532141775513783785693, −1.30779693205822358844244853434, 0,
1.30779693205822358844244853434, 2.22483963532141775513783785693, 3.26702957402052559665502362400, 4.48080927891411339649043282240, 5.21581303955537840355157129629, 5.53623218973341603191591242782, 6.38664649252791786925743548227, 7.03249359501250263096101255518, 7.68158474603930669465349935865
