L(s) = 1 | − 2·3-s − 7-s + 9-s + 11-s − 4·13-s + 4·19-s + 2·21-s + 4·27-s + 6·29-s − 10·31-s − 2·33-s + 2·37-s + 8·39-s − 12·41-s − 4·43-s − 6·47-s + 49-s − 6·53-s − 8·57-s + 6·59-s + 4·61-s − 63-s − 4·67-s + 12·71-s + 4·73-s − 77-s + 8·79-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.917·19-s + 0.436·21-s + 0.769·27-s + 1.11·29-s − 1.79·31-s − 0.348·33-s + 0.328·37-s + 1.28·39-s − 1.87·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.05·57-s + 0.781·59-s + 0.512·61-s − 0.125·63-s − 0.488·67-s + 1.42·71-s + 0.468·73-s − 0.113·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7139771896\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7139771896\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.51038572350914, −12.92144611948434, −12.41046982956068, −12.06532058523125, −11.65907009758317, −11.23016524542968, −10.67179763876004, −10.15492482187584, −9.762415395579515, −9.281643919886236, −8.684967946132057, −8.053384693241728, −7.499796262534142, −6.912293156822965, −6.551680870530283, −6.087953699538275, −5.328561458464249, −5.021685820338400, −4.714457196255711, −3.620483627783119, −3.397894662651076, −2.525316303593580, −1.868976729329513, −1.032016875525754, −0.3175339821148610,
0.3175339821148610, 1.032016875525754, 1.868976729329513, 2.525316303593580, 3.397894662651076, 3.620483627783119, 4.714457196255711, 5.021685820338400, 5.328561458464249, 6.087953699538275, 6.551680870530283, 6.912293156822965, 7.499796262534142, 8.053384693241728, 8.684967946132057, 9.281643919886236, 9.762415395579515, 10.15492482187584, 10.67179763876004, 11.23016524542968, 11.65907009758317, 12.06532058523125, 12.41046982956068, 12.92144611948434, 13.51038572350914