| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 13-s − 15-s + 6·17-s + 2·21-s + 25-s + 27-s − 8·29-s − 2·31-s − 2·35-s + 4·37-s + 39-s − 10·41-s − 4·43-s − 45-s − 3·49-s + 6·51-s + 8·53-s − 14·61-s + 2·63-s − 65-s + 4·67-s − 16·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 1.45·17-s + 0.436·21-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.359·31-s − 0.338·35-s + 0.657·37-s + 0.160·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 3/7·49-s + 0.840·51-s + 1.09·53-s − 1.79·61-s + 0.251·63-s − 0.124·65-s + 0.488·67-s − 1.89·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.039978386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.039978386\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.17545586095280, −12.66557994924365, −12.10799629107915, −11.75618630689906, −11.27223109464642, −10.79848816420349, −10.24738624430568, −9.834802561740022, −9.229155743516807, −8.841908490018833, −8.196474901717245, −7.933297100971394, −7.467453475242205, −7.058682939361974, −6.317913434166740, −5.780163152782495, −5.147800774922653, −4.841823023478253, −4.037326762461997, −3.616792769619442, −3.202634314126751, −2.488077051072195, −1.696046708939469, −1.390626930488922, −0.4760976785782681,
0.4760976785782681, 1.390626930488922, 1.696046708939469, 2.488077051072195, 3.202634314126751, 3.616792769619442, 4.037326762461997, 4.841823023478253, 5.147800774922653, 5.780163152782495, 6.317913434166740, 7.058682939361974, 7.467453475242205, 7.933297100971394, 8.196474901717245, 8.841908490018833, 9.229155743516807, 9.834802561740022, 10.24738624430568, 10.79848816420349, 11.27223109464642, 11.75618630689906, 12.10799629107915, 12.66557994924365, 13.17545586095280
