| L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s − 6·13-s − 2·15-s − 2·17-s + 6·19-s − 4·21-s − 4·23-s + 25-s + 4·27-s − 2·29-s − 6·31-s + 2·35-s + 37-s + 12·39-s − 6·41-s + 8·43-s + 45-s − 6·47-s − 3·49-s + 4·51-s − 10·53-s − 12·57-s + 14·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.485·17-s + 1.37·19-s − 0.872·21-s − 0.834·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s − 1.07·31-s + 0.338·35-s + 0.164·37-s + 1.92·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.560·51-s − 1.37·53-s − 1.58·57-s + 1.82·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.347853271040922321452254411051, −8.178957074481073668057998940629, −7.35951337351047442755521078002, −6.62480561254250018150161345212, −5.47521400662582269779531059657, −5.24711603893625671187585278140, −4.27942991552327182059098455072, −2.76432837165668816106238211903, −1.59039661464182765903059953927, 0,
1.59039661464182765903059953927, 2.76432837165668816106238211903, 4.27942991552327182059098455072, 5.24711603893625671187585278140, 5.47521400662582269779531059657, 6.62480561254250018150161345212, 7.35951337351047442755521078002, 8.178957074481073668057998940629, 9.347853271040922321452254411051
