| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 4·11-s − 3·13-s − 14-s + 16-s + 7·17-s + 2·19-s + 4·22-s + 23-s + 3·26-s + 28-s + 29-s − 9·31-s − 32-s − 7·34-s − 2·37-s − 2·38-s + 6·41-s − 11·43-s − 4·44-s − 46-s + 6·47-s + 49-s − 3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.20·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s + 1.69·17-s + 0.458·19-s + 0.852·22-s + 0.208·23-s + 0.588·26-s + 0.188·28-s + 0.185·29-s − 1.61·31-s − 0.176·32-s − 1.20·34-s − 0.328·37-s − 0.324·38-s + 0.937·41-s − 1.67·43-s − 0.603·44-s − 0.147·46-s + 0.875·47-s + 1/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−7.55928860605739154210698222487, −7.01498688715390618905279228044, −5.86621327930023070124448051376, −5.39168104094921878834494006633, −4.78794971320813223384508591361, −3.60888412405831698425495005593, −2.91043443988986850007164776957, −2.10344326037713986284896925993, −1.14381762831394285871366878787, 0,
1.14381762831394285871366878787, 2.10344326037713986284896925993, 2.91043443988986850007164776957, 3.60888412405831698425495005593, 4.78794971320813223384508591361, 5.39168104094921878834494006633, 5.86621327930023070124448051376, 7.01498688715390618905279228044, 7.55928860605739154210698222487
