| L(s) = 1 | + 0.819·2-s + 3-s − 1.32·4-s + 0.332·5-s + 0.819·6-s − 2.72·8-s + 9-s + 0.272·10-s + 1.66·11-s − 1.32·12-s + 0.0164·13-s + 0.332·15-s + 0.418·16-s + 1.78·17-s + 0.819·18-s − 2.53·19-s − 0.440·20-s + 1.36·22-s − 2.87·23-s − 2.72·24-s − 4.88·25-s + 0.0134·26-s + 27-s + 0.605·29-s + 0.272·30-s − 7.60·31-s + 5.79·32-s + ⋯ |
| L(s) = 1 | + 0.579·2-s + 0.577·3-s − 0.663·4-s + 0.148·5-s + 0.334·6-s − 0.964·8-s + 0.333·9-s + 0.0861·10-s + 0.501·11-s − 0.383·12-s + 0.00456·13-s + 0.0857·15-s + 0.104·16-s + 0.433·17-s + 0.193·18-s − 0.581·19-s − 0.0985·20-s + 0.290·22-s − 0.599·23-s − 0.556·24-s − 0.977·25-s + 0.00264·26-s + 0.192·27-s + 0.112·29-s + 0.0497·30-s − 1.36·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.819T + 2T^{2} \) |
| 5 | \( 1 - 0.332T + 5T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 - 0.0164T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 0.605T + 29T^{2} \) |
| 31 | \( 1 + 7.60T + 31T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 - 3.84T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 4.42T + 61T^{2} \) |
| 67 | \( 1 - 4.72T + 67T^{2} \) |
| 71 | \( 1 - 5.80T + 71T^{2} \) |
| 73 | \( 1 - 0.748T + 73T^{2} \) |
| 79 | \( 1 - 4.49T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 5.00T + 89T^{2} \) |
| 97 | \( 1 - 6.40T + 97T^{2} \) |
| 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
−7.962570478826541315728576221715, −6.90990908893516110130108206822, −6.22069169641845066998742506141, −5.46909613728436582338164418642, −4.77445254160006032728785749265, −3.84523721789232339561250690319, −3.56849825483040451970113021035, −2.45632415298573175529501137685, −1.48849318474692407283212431796, 0,
1.48849318474692407283212431796, 2.45632415298573175529501137685, 3.56849825483040451970113021035, 3.84523721789232339561250690319, 4.77445254160006032728785749265, 5.46909613728436582338164418642, 6.22069169641845066998742506141, 6.90990908893516110130108206822, 7.962570478826541315728576221715
