L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s − 6·13-s − 6·17-s + 6·19-s + 21-s − 23-s − 5·25-s + 27-s + 2·29-s − 8·31-s + 2·33-s + 8·37-s − 6·39-s + 6·41-s − 10·43-s + 8·47-s + 49-s − 6·51-s − 4·53-s + 6·57-s − 8·59-s + 63-s − 14·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 1.45·17-s + 1.37·19-s + 0.218·21-s − 0.208·23-s − 25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.348·33-s + 1.31·37-s − 0.960·39-s + 0.937·41-s − 1.52·43-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.549·53-s + 0.794·57-s − 1.04·59-s + 0.125·63-s − 1.71·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.49098511128869798518007552948, −7.08004359951685943646394397625, −6.16656280762608211112739288795, −5.33356479530101211629066186295, −4.56230056726952001611077983786, −4.02984889654972318502823312101, −2.98542287811092465891826173635, −2.29070519780196360652996988825, −1.46147562329921933909172986719, 0,
1.46147562329921933909172986719, 2.29070519780196360652996988825, 2.98542287811092465891826173635, 4.02984889654972318502823312101, 4.56230056726952001611077983786, 5.33356479530101211629066186295, 6.16656280762608211112739288795, 7.08004359951685943646394397625, 7.49098511128869798518007552948