| L(s) = 1 | − 2·4-s + 5-s − 13-s + 4·16-s − 6·17-s + 5·19-s − 2·20-s − 6·23-s + 25-s + 6·29-s + 5·31-s − 7·37-s − 12·41-s − 43-s − 6·47-s + 2·52-s + 6·59-s + 2·61-s − 8·64-s − 65-s − 7·67-s + 12·68-s − 12·71-s + 11·73-s − 10·76-s − 13·79-s + 4·80-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.277·13-s + 16-s − 1.45·17-s + 1.14·19-s − 0.447·20-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.898·31-s − 1.15·37-s − 1.87·41-s − 0.152·43-s − 0.875·47-s + 0.277·52-s + 0.781·59-s + 0.256·61-s − 64-s − 0.124·65-s − 0.855·67-s + 1.45·68-s − 1.42·71-s + 1.28·73-s − 1.14·76-s − 1.46·79-s + 0.447·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.590875702685233331934022648353, −8.197502747986474174878327685627, −7.04645975965922436536349711500, −6.30756202040947465508996354014, −5.30761125151519086066342212648, −4.72972591721354629179262187637, −3.82855621842570259866356112592, −2.77119704212117661375561446938, −1.50947380544378198966242198049, 0,
1.50947380544378198966242198049, 2.77119704212117661375561446938, 3.82855621842570259866356112592, 4.72972591721354629179262187637, 5.30761125151519086066342212648, 6.30756202040947465508996354014, 7.04645975965922436536349711500, 8.197502747986474174878327685627, 8.590875702685233331934022648353
