| L(s) = 1 | + 7-s − 2·11-s + 6·13-s − 19-s − 6·23-s − 5·25-s − 6·29-s + 8·31-s + 6·37-s − 6·41-s − 12·43-s + 6·47-s + 49-s + 2·53-s − 12·59-s + 2·61-s − 8·67-s − 12·71-s − 6·73-s − 2·77-s + 14·83-s + 2·89-s + 6·91-s − 6·97-s + 12·101-s + 4·103-s − 20·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.603·11-s + 1.66·13-s − 0.229·19-s − 1.25·23-s − 25-s − 1.11·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s − 1.82·43-s + 0.875·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s + 0.256·61-s − 0.977·67-s − 1.42·71-s − 0.702·73-s − 0.227·77-s + 1.53·83-s + 0.211·89-s + 0.628·91-s − 0.609·97-s + 1.19·101-s + 0.394·103-s − 1.93·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.57934239798453907586523220666, −6.45235162821108458444526668429, −6.06584993707892060052438867793, −5.38523198983913634854509715500, −4.47055775579774275837323130538, −3.86887918606260545432781516059, −3.09310493128193359645133490192, −2.06478025261328066094313941397, −1.33087661064582686941609043603, 0,
1.33087661064582686941609043603, 2.06478025261328066094313941397, 3.09310493128193359645133490192, 3.86887918606260545432781516059, 4.47055775579774275837323130538, 5.38523198983913634854509715500, 6.06584993707892060052438867793, 6.45235162821108458444526668429, 7.57934239798453907586523220666
