| L(s) = 1 | − 3·3-s − 7-s + 6·9-s − 3·13-s − 17-s − 6·19-s + 3·21-s − 6·23-s − 9·27-s + 9·29-s − 4·31-s − 2·37-s + 9·39-s + 4·41-s + 10·43-s + 47-s + 49-s + 3·51-s − 4·53-s + 18·57-s − 8·59-s + 8·61-s − 6·63-s − 12·67-s + 18·69-s + 8·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s − 0.832·13-s − 0.242·17-s − 1.37·19-s + 0.654·21-s − 1.25·23-s − 1.73·27-s + 1.67·29-s − 0.718·31-s − 0.328·37-s + 1.44·39-s + 0.624·41-s + 1.52·43-s + 0.145·47-s + 1/7·49-s + 0.420·51-s − 0.549·53-s + 2.38·57-s − 1.04·59-s + 1.02·61-s − 0.755·63-s − 1.46·67-s + 2.16·69-s + 0.949·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.35003924168657, −13.49573884441273, −13.02825628555173, −12.40932036641062, −12.24214115905402, −11.88019580520343, −11.05222415179786, −10.80668854142899, −10.34153964364232, −9.864016068260404, −9.358273903023234, −8.662385392206374, −8.019581034170508, −7.398050553767886, −6.870796368311143, −6.377474060478162, −6.002155777135784, −5.490317640779824, −4.840744170261865, −4.295162994545298, −4.018954355325759, −2.905541217800880, −2.252345849207311, −1.507854629786239, −0.5928627986099969, 0,
0.5928627986099969, 1.507854629786239, 2.252345849207311, 2.905541217800880, 4.018954355325759, 4.295162994545298, 4.840744170261865, 5.490317640779824, 6.002155777135784, 6.377474060478162, 6.870796368311143, 7.398050553767886, 8.019581034170508, 8.662385392206374, 9.358273903023234, 9.864016068260404, 10.34153964364232, 10.80668854142899, 11.05222415179786, 11.88019580520343, 12.24214115905402, 12.40932036641062, 13.02825628555173, 13.49573884441273, 14.35003924168657
