| L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 9-s + 4·11-s + 6·13-s + 4·15-s + 4·17-s − 4·19-s − 2·21-s − 25-s − 4·27-s + 10·29-s − 2·31-s + 8·33-s − 2·35-s + 12·39-s − 2·41-s + 4·43-s + 2·45-s + 49-s + 8·51-s + 6·53-s + 8·55-s − 8·57-s + 14·61-s − 63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 0.436·21-s − 1/5·25-s − 0.769·27-s + 1.85·29-s − 0.359·31-s + 1.39·33-s − 0.338·35-s + 1.92·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 1.07·55-s − 1.05·57-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.353877434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.353877434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.11628878401155, −13.53794206251459, −13.28050610817417, −12.75880903543865, −12.04594558175807, −11.62800648615292, −10.98280188707268, −10.32845434693108, −9.945001989093984, −9.456473148649747, −8.841164422213017, −8.567727153989461, −8.230563374529070, −7.381708687704985, −6.777682248944444, −6.201197743639283, −5.933875683741462, −5.270472982900398, −4.221333909523781, −3.904052484611629, −3.309427200264676, −2.746296081571402, −2.040435205645041, −1.438118329627934, −0.8052891854472825,
0.8052891854472825, 1.438118329627934, 2.040435205645041, 2.746296081571402, 3.309427200264676, 3.904052484611629, 4.221333909523781, 5.270472982900398, 5.933875683741462, 6.201197743639283, 6.777682248944444, 7.381708687704985, 8.230563374529070, 8.567727153989461, 8.841164422213017, 9.456473148649747, 9.945001989093984, 10.32845434693108, 10.98280188707268, 11.62800648615292, 12.04594558175807, 12.75880903543865, 13.28050610817417, 13.53794206251459, 14.11628878401155
