| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.384716132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.384716132\) |
| \(L(\frac{3}{2})\) |
|
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
−13.95230751207737, −13.49315380133422, −13.07701413364390, −12.52114486744633, −12.13025378809754, −11.43752095841343, −10.98411229819020, −10.50365139536634, −9.973816539572452, −9.470163160409858, −9.107016711383828, −8.402074128862591, −7.734274421880387, −7.482814033824121, −6.913025953869330, −6.103220246295859, −5.706423638141940, −5.093930980175111, −4.766589698582666, −3.917326510429364, −3.196542113174415, −2.684011009793633, −2.136816759858057, −1.250440889432108, −0.5274584308394039,
0.5274584308394039, 1.250440889432108, 2.136816759858057, 2.684011009793633, 3.196542113174415, 3.917326510429364, 4.766589698582666, 5.093930980175111, 5.706423638141940, 6.103220246295859, 6.913025953869330, 7.482814033824121, 7.734274421880387, 8.402074128862591, 9.107016711383828, 9.470163160409858, 9.973816539572452, 10.50365139536634, 10.98411229819020, 11.43752095841343, 12.13025378809754, 12.52114486744633, 13.07701413364390, 13.49315380133422, 13.95230751207737
