| L(s) = 1 | + 7-s − 11-s + 4·13-s + 4·19-s + 6·29-s + 10·31-s − 2·37-s + 12·41-s − 4·43-s − 6·47-s + 49-s − 6·53-s − 6·59-s − 4·61-s − 4·67-s + 12·71-s + 4·73-s − 77-s − 8·79-s − 12·83-s − 18·89-s + 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.301·11-s + 1.10·13-s + 0.917·19-s + 1.11·29-s + 1.79·31-s − 0.328·37-s + 1.87·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 0.781·59-s − 0.512·61-s − 0.488·67-s + 1.42·71-s + 0.468·73-s − 0.113·77-s − 0.900·79-s − 1.31·83-s − 1.90·89-s + 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.497759605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.497759605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 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
−12.76833228659425, −12.28448973470783, −11.75671831815001, −11.44278153387281, −10.87803248798485, −10.58778242131844, −9.925994613203164, −9.617974024234769, −9.048592178341286, −8.421022989544511, −8.165803068766566, −7.776222662861530, −7.069062394965395, −6.656577375162218, −6.038429404440700, −5.769016231810187, −5.041972347806140, −4.571024102020823, −4.209348009536883, −3.379335742337413, −3.012224790432997, −2.477977949334665, −1.613867842606689, −1.160596960383085, −0.5498779613522048,
0.5498779613522048, 1.160596960383085, 1.613867842606689, 2.477977949334665, 3.012224790432997, 3.379335742337413, 4.209348009536883, 4.571024102020823, 5.041972347806140, 5.769016231810187, 6.038429404440700, 6.656577375162218, 7.069062394965395, 7.776222662861530, 8.165803068766566, 8.421022989544511, 9.048592178341286, 9.617974024234769, 9.925994613203164, 10.58778242131844, 10.87803248798485, 11.44278153387281, 11.75671831815001, 12.28448973470783, 12.76833228659425
