| L(s) = 1 | + 5-s + 4·11-s − 13-s − 17-s − 8·19-s + 25-s + 6·29-s + 8·31-s + 2·37-s + 10·41-s − 4·43-s − 7·49-s + 6·53-s + 4·55-s − 6·61-s − 65-s − 4·67-s − 8·71-s − 14·73-s − 4·79-s − 4·83-s − 85-s + 14·89-s − 8·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.277·13-s − 0.242·17-s − 1.83·19-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 1.56·41-s − 0.609·43-s − 49-s + 0.824·53-s + 0.539·55-s − 0.768·61-s − 0.124·65-s − 0.488·67-s − 0.949·71-s − 1.63·73-s − 0.450·79-s − 0.439·83-s − 0.108·85-s + 1.48·89-s − 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.52296438745977, −13.04501281961337, −12.57069711949404, −12.12654316061507, −11.61871200165968, −11.22873563649637, −10.53628120387301, −10.21289305377565, −9.768140872288996, −9.095011632646270, −8.763819216008309, −8.367671790978783, −7.714968040798082, −7.089267488851234, −6.574357787914665, −6.110706847801049, −5.963735185724506, −4.893981353831997, −4.483354421010345, −4.197331723154569, −3.383359035622300, −2.704891114838570, −2.246310946409142, −1.509005405791467, −0.9379917410196654, 0,
0.9379917410196654, 1.509005405791467, 2.246310946409142, 2.704891114838570, 3.383359035622300, 4.197331723154569, 4.483354421010345, 4.893981353831997, 5.963735185724506, 6.110706847801049, 6.574357787914665, 7.089267488851234, 7.714968040798082, 8.367671790978783, 8.763819216008309, 9.095011632646270, 9.768140872288996, 10.21289305377565, 10.53628120387301, 11.22873563649637, 11.61871200165968, 12.12654316061507, 12.57069711949404, 13.04501281961337, 13.52296438745977
