| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 6·11-s + 2·13-s − 15-s − 8·17-s − 6·19-s − 4·21-s + 4·23-s + 25-s − 27-s + 2·29-s − 4·31-s − 6·33-s + 4·35-s − 8·37-s − 2·39-s + 6·41-s + 43-s + 45-s − 12·47-s + 9·49-s + 8·51-s + 6·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s − 0.258·15-s − 1.94·17-s − 1.37·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 1.04·33-s + 0.676·35-s − 1.31·37-s − 0.320·39-s + 0.937·41-s + 0.152·43-s + 0.149·45-s − 1.75·47-s + 9/7·49-s + 1.12·51-s + 0.809·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.95909158385243, −14.59980001791111, −13.90014430118641, −13.60182846484084, −12.86117301128762, −12.40536190081636, −11.71078226007666, −11.31117669379564, −10.80304840427962, −10.71707040565497, −9.662679549449280, −8.995055221006327, −8.771624425257553, −8.279423664941658, −7.335616059350524, −6.799256391964080, −6.365158020342167, −5.888056173609076, −5.019623126757984, −4.442812073849978, −4.269353608113346, −3.335765731477428, −2.212270608491614, −1.661026416767821, −1.244949645826656, 0,
1.244949645826656, 1.661026416767821, 2.212270608491614, 3.335765731477428, 4.269353608113346, 4.442812073849978, 5.019623126757984, 5.888056173609076, 6.365158020342167, 6.799256391964080, 7.335616059350524, 8.279423664941658, 8.771624425257553, 8.995055221006327, 9.662679549449280, 10.71707040565497, 10.80304840427962, 11.31117669379564, 11.71078226007666, 12.40536190081636, 12.86117301128762, 13.60182846484084, 13.90014430118641, 14.59980001791111, 14.95909158385243
