| L(s) = 1 | − 3-s − 2·5-s + 7-s − 2·9-s + 11-s − 6·13-s + 2·15-s − 4·17-s − 8·19-s − 21-s + 6·23-s − 25-s + 5·27-s + 2·29-s − 4·31-s − 33-s − 2·35-s − 37-s + 6·39-s + 7·41-s + 2·43-s + 4·45-s + 9·47-s − 6·49-s + 4·51-s − 3·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 1.66·13-s + 0.516·15-s − 0.970·17-s − 1.83·19-s − 0.218·21-s + 1.25·23-s − 1/5·25-s + 0.962·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s − 0.338·35-s − 0.164·37-s + 0.960·39-s + 1.09·41-s + 0.304·43-s + 0.596·45-s + 1.31·47-s − 6/7·49-s + 0.560·51-s − 0.412·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−11.19524954917795404395655592484, −10.78391283975406205668829665843, −9.314280646871158472315859842620, −8.404519639968086535147674204319, −7.36881114003389544173968798730, −6.40622823243653800847643763088, −5.06148787959059824431190870012, −4.21402845745683233262500412192, −2.50673634667839885265978774188, 0,
2.50673634667839885265978774188, 4.21402845745683233262500412192, 5.06148787959059824431190870012, 6.40622823243653800847643763088, 7.36881114003389544173968798730, 8.404519639968086535147674204319, 9.314280646871158472315859842620, 10.78391283975406205668829665843, 11.19524954917795404395655592484
