| L(s) = 1 | − 2-s − 0.445·3-s + 4-s + 0.753·5-s + 0.445·6-s − 1.60·7-s − 8-s − 2.80·9-s − 0.753·10-s − 3.60·11-s − 0.445·12-s + 1.60·14-s − 0.335·15-s + 16-s − 1.40·17-s + 2.80·18-s − 19-s + 0.753·20-s + 0.713·21-s + 3.60·22-s − 8.59·23-s + 0.445·24-s − 4.43·25-s + 2.58·27-s − 1.60·28-s − 1.38·29-s + 0.335·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.256·3-s + 0.5·4-s + 0.336·5-s + 0.181·6-s − 0.606·7-s − 0.353·8-s − 0.933·9-s − 0.238·10-s − 1.08·11-s − 0.128·12-s + 0.428·14-s − 0.0865·15-s + 0.250·16-s − 0.340·17-s + 0.660·18-s − 0.229·19-s + 0.168·20-s + 0.155·21-s + 0.768·22-s − 1.79·23-s + 0.0908·24-s − 0.886·25-s + 0.496·27-s − 0.303·28-s − 0.257·29-s + 0.0611·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3891246952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3891246952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.445T + 3T^{2} \) |
| 5 | \( 1 - 0.753T + 5T^{2} \) |
| 7 | \( 1 + 1.60T + 7T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 23 | \( 1 + 8.59T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 - 1.32T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 - 0.219T + 47T^{2} \) |
| 53 | \( 1 + 5.50T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 2.66T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 1.40T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 3.10T + 83T^{2} \) |
| 89 | \( 1 + 2.19T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.912319902498915708627680453970, −7.62975890065188176774797410668, −6.37620833273903950312954745428, −6.12139456677838306510917710531, −5.42537299189259288773018379559, −4.45528246335055739215434978239, −3.38268243412735518275813664820, −2.58838842449701138410649643717, −1.88472552262529936655715397395, −0.33979266278990368143120891231,
0.33979266278990368143120891231, 1.88472552262529936655715397395, 2.58838842449701138410649643717, 3.38268243412735518275813664820, 4.45528246335055739215434978239, 5.42537299189259288773018379559, 6.12139456677838306510917710531, 6.37620833273903950312954745428, 7.62975890065188176774797410668, 7.912319902498915708627680453970
