| L(s) = 1 | − 3-s − 5-s − 2·9-s − 3·11-s − 6·13-s + 15-s + 5·17-s + 19-s − 7·23-s − 4·25-s + 5·27-s − 2·29-s + 5·31-s + 3·33-s − 3·37-s + 6·39-s + 2·41-s + 4·43-s + 2·45-s − 5·47-s − 5·51-s + 53-s + 3·55-s − 57-s + 15·59-s − 5·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 1.66·13-s + 0.258·15-s + 1.21·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.962·27-s − 0.371·29-s + 0.898·31-s + 0.522·33-s − 0.493·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s − 0.729·47-s − 0.700·51-s + 0.137·53-s + 0.404·55-s − 0.132·57-s + 1.95·59-s − 0.640·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6379074933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6379074933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.477656549727874393422066847084, −7.82830516711369004018271381451, −7.39447814586933003973621730528, −6.31319457642689362922902445523, −5.50124088065579204535865736120, −5.06511071409866508052902087436, −4.04159955816643176035584796125, −3.00645085864571493614343512088, −2.17025746450353686276979292547, −0.46739477050181469477720707475,
0.46739477050181469477720707475, 2.17025746450353686276979292547, 3.00645085864571493614343512088, 4.04159955816643176035584796125, 5.06511071409866508052902087436, 5.50124088065579204535865736120, 6.31319457642689362922902445523, 7.39447814586933003973621730528, 7.82830516711369004018271381451, 8.477656549727874393422066847084
