L(s) = 1 | − 2·5-s + 11-s + 2·13-s − 2·17-s + 8·23-s − 25-s + 6·29-s − 8·31-s + 6·37-s − 2·41-s − 8·47-s − 6·53-s − 2·55-s + 4·59-s − 6·61-s − 4·65-s + 4·67-s + 14·73-s + 4·79-s − 12·83-s + 4·85-s − 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.301·11-s + 0.554·13-s − 0.485·17-s + 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s − 1.16·47-s − 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + 1.63·73-s + 0.450·79-s − 1.31·83-s + 0.433·85-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.37359126406776, −13.69467510822570, −13.31168896764603, −12.60331925944546, −12.48611511555815, −11.61592359386579, −11.23960435057659, −11.02788170655081, −10.37573893895687, −9.589107737037921, −9.298588032516067, −8.531640469877918, −8.321750575987015, −7.616330696774009, −7.120038147034913, −6.608184818294366, −6.107441291105620, −5.325506122645638, −4.785075694495303, −4.258794806091169, −3.599754675031618, −3.187523071190326, −2.431858954388236, −1.561781723360506, −0.8757656668354690, 0,
0.8757656668354690, 1.561781723360506, 2.431858954388236, 3.187523071190326, 3.599754675031618, 4.258794806091169, 4.785075694495303, 5.325506122645638, 6.107441291105620, 6.608184818294366, 7.120038147034913, 7.616330696774009, 8.321750575987015, 8.531640469877918, 9.298588032516067, 9.589107737037921, 10.37573893895687, 11.02788170655081, 11.23960435057659, 11.61592359386579, 12.48611511555815, 12.60331925944546, 13.31168896764603, 13.69467510822570, 14.37359126406776