| L(s) = 1 | − 2-s + 4-s − 8-s − 4·11-s − 2·13-s + 16-s − 4·17-s + 4·22-s − 6·23-s + 2·26-s − 10·29-s − 2·31-s − 32-s + 4·34-s − 2·37-s − 6·41-s + 8·43-s − 4·44-s + 6·46-s + 6·47-s − 7·49-s − 2·52-s − 6·53-s + 10·58-s − 4·59-s + 6·61-s + 2·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.970·17-s + 0.852·22-s − 1.25·23-s + 0.392·26-s − 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s − 0.328·37-s − 0.937·41-s + 1.21·43-s − 0.603·44-s + 0.884·46-s + 0.875·47-s − 49-s − 0.277·52-s − 0.824·53-s + 1.31·58-s − 0.520·59-s + 0.768·61-s + 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.37381955435867, −13.06194041555767, −12.46032026239782, −12.11989527989192, −11.46672862309933, −10.92622246633918, −10.76596420453551, −10.11213289803450, −9.611625137168731, −9.327133604578337, −8.641940910428253, −8.212449026059614, −7.679149317896778, −7.360851275798522, −6.817612268638594, −6.148051343845928, −5.693368231170272, −5.160241602316727, −4.590444811847044, −3.905025777340690, −3.342789882405404, −2.596176688289689, −2.076172708993685, −1.725222848634504, −0.5391839654263019, 0,
0.5391839654263019, 1.725222848634504, 2.076172708993685, 2.596176688289689, 3.342789882405404, 3.905025777340690, 4.590444811847044, 5.160241602316727, 5.693368231170272, 6.148051343845928, 6.817612268638594, 7.360851275798522, 7.679149317896778, 8.212449026059614, 8.641940910428253, 9.327133604578337, 9.611625137168731, 10.11213289803450, 10.76596420453551, 10.92622246633918, 11.46672862309933, 12.11989527989192, 12.46032026239782, 13.06194041555767, 13.37381955435867
