| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s + 13-s + 16-s + 2·19-s + 20-s + 4·22-s − 4·23-s + 25-s − 26-s − 6·29-s − 32-s − 2·38-s − 40-s − 6·41-s − 10·43-s − 4·44-s + 4·46-s − 7·49-s − 50-s + 52-s − 12·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.458·19-s + 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s − 0.196·26-s − 1.11·29-s − 0.176·32-s − 0.324·38-s − 0.158·40-s − 0.937·41-s − 1.52·43-s − 0.603·44-s + 0.589·46-s − 49-s − 0.141·50-s + 0.138·52-s − 1.64·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.15265500040287, −12.51801723945230, −12.06296985949050, −11.62392295052025, −11.04488261930464, −10.75912776419261, −10.22322619119637, −9.834632522967921, −9.492095642187820, −8.967525046519931, −8.351205380057348, −8.066924816166455, −7.589844836062284, −7.167478018197975, −6.468303103950802, −6.165069067841226, −5.594707510491214, −5.056238254440880, −4.747021532029696, −3.831549428481180, −3.311393080051156, −2.878755399451382, −2.154518644612191, −1.726720355365830, −1.184813215389222, 0, 0,
1.184813215389222, 1.726720355365830, 2.154518644612191, 2.878755399451382, 3.311393080051156, 3.831549428481180, 4.747021532029696, 5.056238254440880, 5.594707510491214, 6.165069067841226, 6.468303103950802, 7.167478018197975, 7.589844836062284, 8.066924816166455, 8.351205380057348, 8.967525046519931, 9.492095642187820, 9.834632522967921, 10.22322619119637, 10.75912776419261, 11.04488261930464, 11.62392295052025, 12.06296985949050, 12.51801723945230, 13.15265500040287
