| L(s) = 1 | − 2·3-s + 7-s + 9-s − 11-s + 13-s + 6·17-s − 4·19-s − 2·21-s − 5·25-s + 4·27-s + 6·29-s − 4·31-s + 2·33-s + 2·37-s − 2·39-s − 6·41-s + 2·43-s + 49-s − 12·51-s + 6·53-s + 8·57-s − 10·61-s + 63-s − 4·67-s + 14·73-s + 10·75-s − 77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 1.45·17-s − 0.917·19-s − 0.436·21-s − 25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.304·43-s + 1/7·49-s − 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.28·61-s + 0.125·63-s − 0.488·67-s + 1.63·73-s + 1.15·75-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.077574748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.077574748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.299487922797766965496615811313, −7.75963238433379083612323349093, −6.83427582813446212150490330257, −6.08526002195855766499070044041, −5.54459018511350528891598841942, −4.87264221582484270767013963939, −4.00417892635685847250300799876, −2.98699747544327276720351903158, −1.77141147281134253935723284878, −0.63824049979866607471288003490,
0.63824049979866607471288003490, 1.77141147281134253935723284878, 2.98699747544327276720351903158, 4.00417892635685847250300799876, 4.87264221582484270767013963939, 5.54459018511350528891598841942, 6.08526002195855766499070044041, 6.83427582813446212150490330257, 7.75963238433379083612323349093, 8.299487922797766965496615811313
