| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 13-s − 14-s + 16-s + 4·19-s + 20-s − 3·23-s − 4·25-s − 26-s + 28-s − 6·29-s − 6·31-s − 32-s + 35-s − 12·37-s − 4·38-s − 40-s − 6·41-s + 12·43-s + 3·46-s + 10·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.917·19-s + 0.223·20-s − 0.625·23-s − 4/5·25-s − 0.196·26-s + 0.188·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s + 0.169·35-s − 1.97·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 1.82·43-s + 0.442·46-s + 1.45·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.379279786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.379279786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.96810364403361, −14.32375465800058, −13.85712322596268, −13.49559140096003, −12.72863213031155, −12.00806182136662, −11.90451255747768, −10.89814733407264, −10.77797352986118, −10.11989665476518, −9.389007320930564, −9.170970692867530, −8.560706553210162, −7.767556199824091, −7.477759584987116, −6.904466488357347, −5.997417135312510, −5.697871764981533, −5.094419894925296, −4.154922590956872, −3.558605502157929, −2.825791860731785, −1.818215775203771, −1.655538745540544, −0.4761643390441667,
0.4761643390441667, 1.655538745540544, 1.818215775203771, 2.825791860731785, 3.558605502157929, 4.154922590956872, 5.094419894925296, 5.697871764981533, 5.997417135312510, 6.904466488357347, 7.477759584987116, 7.767556199824091, 8.560706553210162, 9.170970692867530, 9.389007320930564, 10.11989665476518, 10.77797352986118, 10.89814733407264, 11.90451255747768, 12.00806182136662, 12.72863213031155, 13.49559140096003, 13.85712322596268, 14.32375465800058, 14.96810364403361
