| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 4·11-s + 12·12-s + 42·13-s − 14·14-s + 16·16-s + 86·17-s − 18·18-s − 96·19-s + 21·21-s + 8·22-s + 96·23-s − 24·24-s − 84·26-s + 27·27-s + 28·28-s − 78·29-s + 80·31-s − 32·32-s − 12·33-s − 172·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.109·11-s + 0.288·12-s + 0.896·13-s − 0.267·14-s + 1/4·16-s + 1.22·17-s − 0.235·18-s − 1.15·19-s + 0.218·21-s + 0.0775·22-s + 0.870·23-s − 0.204·24-s − 0.633·26-s + 0.192·27-s + 0.188·28-s − 0.499·29-s + 0.463·31-s − 0.176·32-s − 0.0633·33-s − 0.867·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.137820342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137820342\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 86 T + p^{3} T^{2} \) |
| 19 | \( 1 + 96 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 80 T + p^{3} T^{2} \) |
| 37 | \( 1 + 50 T + p^{3} T^{2} \) |
| 41 | \( 1 + 26 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 20 T + p^{3} T^{2} \) |
| 53 | \( 1 - 382 T + p^{3} T^{2} \) |
| 59 | \( 1 - 356 T + p^{3} T^{2} \) |
| 61 | \( 1 + 134 T + p^{3} T^{2} \) |
| 67 | \( 1 + 888 T + p^{3} T^{2} \) |
| 71 | \( 1 - 868 T + p^{3} T^{2} \) |
| 73 | \( 1 - 70 T + p^{3} T^{2} \) |
| 79 | \( 1 - 400 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 + 634 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1202 T + p^{3} T^{2} \) |
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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
−9.395771903767006318519126543806, −8.642778989151379682653236238115, −8.077241683613605270803223589893, −7.25206551250416710294622817443, −6.31863670028366202259990761501, −5.31233461843773949611935119809, −4.05267941580778258953702105809, −3.07106868795656307275054005745, −1.92063361068389450749466021430, −0.867113335632319963965159211147,
0.867113335632319963965159211147, 1.92063361068389450749466021430, 3.07106868795656307275054005745, 4.05267941580778258953702105809, 5.31233461843773949611935119809, 6.31863670028366202259990761501, 7.25206551250416710294622817443, 8.077241683613605270803223589893, 8.642778989151379682653236238115, 9.395771903767006318519126543806
