| L(s) = 1 | − 7-s − 3·9-s − 4·13-s − 4·17-s + 2·19-s − 23-s + 10·29-s + 6·31-s + 6·37-s − 2·41-s − 4·43-s − 10·47-s + 49-s + 10·53-s − 10·61-s + 3·63-s − 12·67-s − 4·71-s + 10·73-s − 8·79-s + 9·81-s + 2·83-s + 4·91-s + 16·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.10·13-s − 0.970·17-s + 0.458·19-s − 0.208·23-s + 1.85·29-s + 1.07·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 1.45·47-s + 1/7·49-s + 1.37·53-s − 1.28·61-s + 0.377·63-s − 1.46·67-s − 0.474·71-s + 1.17·73-s − 0.900·79-s + 81-s + 0.219·83-s + 0.419·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9622299264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9622299264\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.19233963830679, −13.70559436682071, −13.35697550929990, −12.71323070795494, −12.03584283617868, −11.82417734425015, −11.34240888212143, −10.61476076676527, −10.07854500596489, −9.791319575840224, −8.949021217932315, −8.732332247959993, −7.997914152430968, −7.601450019728560, −6.801456626717178, −6.382545907235869, −5.958857744392569, −5.011918466219747, −4.824960252380646, −4.094011655599304, −3.210928132637421, −2.730748166494429, −2.311832109872842, −1.259622602656205, −0.3387076403698521,
0.3387076403698521, 1.259622602656205, 2.311832109872842, 2.730748166494429, 3.210928132637421, 4.094011655599304, 4.824960252380646, 5.011918466219747, 5.958857744392569, 6.382545907235869, 6.801456626717178, 7.601450019728560, 7.997914152430968, 8.732332247959993, 8.949021217932315, 9.791319575840224, 10.07854500596489, 10.61476076676527, 11.34240888212143, 11.82417734425015, 12.03584283617868, 12.71323070795494, 13.35697550929990, 13.70559436682071, 14.19233963830679
