| L(s) = 1 | − 3-s + 5-s + 9-s − 5·11-s − 5·13-s − 15-s − 17-s + 4·19-s + 6·23-s − 4·25-s − 27-s − 2·31-s + 5·33-s − 37-s + 5·39-s − 2·41-s + 43-s + 45-s − 6·47-s + 51-s + 9·53-s − 5·55-s − 4·57-s − 4·59-s + 6·61-s − 5·65-s − 67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.50·11-s − 1.38·13-s − 0.258·15-s − 0.242·17-s + 0.917·19-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 0.359·31-s + 0.870·33-s − 0.164·37-s + 0.800·39-s − 0.312·41-s + 0.152·43-s + 0.149·45-s − 0.875·47-s + 0.140·51-s + 1.23·53-s − 0.674·55-s − 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.620·65-s − 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.33220755558069, −13.02853251868621, −12.68882152941057, −11.96482174808734, −11.76638906837362, −10.98196440090875, −10.78875863670626, −9.971026317883898, −9.940640384265587, −9.365686500571757, −8.740939582249461, −8.121663772310404, −7.486689689189438, −7.335824070031516, −6.705011147631285, −6.089864587692493, −5.438817208608994, −5.071202422587717, −4.935018713999021, −4.040745325486445, −3.339460574480159, −2.658166514980057, −2.305037877590141, −1.554287369296142, −0.6757547947850247, 0,
0.6757547947850247, 1.554287369296142, 2.305037877590141, 2.658166514980057, 3.339460574480159, 4.040745325486445, 4.935018713999021, 5.071202422587717, 5.438817208608994, 6.089864587692493, 6.705011147631285, 7.335824070031516, 7.486689689189438, 8.121663772310404, 8.740939582249461, 9.365686500571757, 9.940640384265587, 9.971026317883898, 10.78875863670626, 10.98196440090875, 11.76638906837362, 11.96482174808734, 12.68882152941057, 13.02853251868621, 13.33220755558069
