| L(s) = 1 | − 5-s − 7-s − 4·13-s − 3·17-s − 8·19-s + 4·23-s − 4·25-s + 8·29-s − 10·31-s + 35-s − 2·37-s + 6·41-s − 43-s + 13·47-s + 49-s + 8·53-s − 3·59-s + 4·65-s + 7·67-s − 6·71-s − 6·73-s + 8·79-s + 83-s + 3·85-s − 3·89-s + 4·91-s + 8·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.10·13-s − 0.727·17-s − 1.83·19-s + 0.834·23-s − 4/5·25-s + 1.48·29-s − 1.79·31-s + 0.169·35-s − 0.328·37-s + 0.937·41-s − 0.152·43-s + 1.89·47-s + 1/7·49-s + 1.09·53-s − 0.390·59-s + 0.496·65-s + 0.855·67-s − 0.712·71-s − 0.702·73-s + 0.900·79-s + 0.109·83-s + 0.325·85-s − 0.317·89-s + 0.419·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.59354601451964, −14.12125462061186, −13.45566106162335, −12.97927733086858, −12.39927819062437, −12.27070675407361, −11.45763262696820, −10.97640002957167, −10.48612609160934, −10.06506794822704, −9.312204636145556, −8.849317371110561, −8.513770021479166, −7.646623580101400, −7.300706586496348, −6.737567879375461, −6.205835195779619, −5.540466634296413, −4.900651208528696, −4.215627314486214, −3.963400518893525, −3.020181314773399, −2.394820227787846, −1.922121602460239, −0.7114064498789214, 0,
0.7114064498789214, 1.922121602460239, 2.394820227787846, 3.020181314773399, 3.963400518893525, 4.215627314486214, 4.900651208528696, 5.540466634296413, 6.205835195779619, 6.737567879375461, 7.300706586496348, 7.646623580101400, 8.513770021479166, 8.849317371110561, 9.312204636145556, 10.06506794822704, 10.48612609160934, 10.97640002957167, 11.45763262696820, 12.27070675407361, 12.39927819062437, 12.97927733086858, 13.45566106162335, 14.12125462061186, 14.59354601451964
