| L(s) = 1 | − 3-s − 5-s + 9-s + 2·11-s − 4·13-s + 15-s + 25-s − 27-s − 2·29-s + 10·31-s − 2·33-s − 4·37-s + 4·39-s + 6·41-s + 4·43-s − 45-s − 7·49-s − 2·53-s − 2·55-s + 6·59-s − 6·61-s + 4·65-s + 4·67-s + 8·71-s + 14·73-s − 75-s + 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.258·15-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.79·31-s − 0.348·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s − 0.274·53-s − 0.269·55-s + 0.781·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.63·73-s − 0.115·75-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165877313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165877313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.354743017063827992703511762583, −8.329582116498729375493628464464, −7.57991237434791376490491311680, −6.81096099259753091104697581374, −6.10069729795284960484591204157, −5.04292717164075000308373592653, −4.42607394051627953525124391189, −3.40131096547709806905683341241, −2.19770710980700157475151472361, −0.74394243212741429594461253345,
0.74394243212741429594461253345, 2.19770710980700157475151472361, 3.40131096547709806905683341241, 4.42607394051627953525124391189, 5.04292717164075000308373592653, 6.10069729795284960484591204157, 6.81096099259753091104697581374, 7.57991237434791376490491311680, 8.329582116498729375493628464464, 9.354743017063827992703511762583
