L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 2·13-s − 15-s − 2·17-s − 2·21-s + 2·23-s + 25-s − 27-s + 4·29-s + 4·31-s + 2·35-s − 2·37-s + 2·39-s + 4·41-s + 10·43-s + 45-s − 6·47-s − 3·49-s + 2·51-s − 6·53-s + 4·59-s + 10·61-s + 2·63-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.338·35-s − 0.328·37-s + 0.320·39-s + 0.624·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 0.251·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.68000980115655, −12.43571027803719, −11.71455970559436, −11.46039757081777, −11.02536872389327, −10.58092332146150, −10.03068396154081, −9.764233956883453, −9.128997621088956, −8.714849979183899, −8.163573015444111, −7.731637117386585, −7.203924072144971, −6.661490592012081, −6.349045038383707, −5.707170547951457, −5.267424900625255, −4.772663128250155, −4.464477502658460, −3.852054838712298, −3.091485998249325, −2.490145645319519, −2.068541943391820, −1.312355471179007, −0.8544247254665940, 0,
0.8544247254665940, 1.312355471179007, 2.068541943391820, 2.490145645319519, 3.091485998249325, 3.852054838712298, 4.464477502658460, 4.772663128250155, 5.267424900625255, 5.707170547951457, 6.349045038383707, 6.661490592012081, 7.203924072144971, 7.731637117386585, 8.163573015444111, 8.714849979183899, 9.128997621088956, 9.764233956883453, 10.03068396154081, 10.58092332146150, 11.02536872389327, 11.46039757081777, 11.71455970559436, 12.43571027803719, 12.68000980115655