| L(s) = 1 | + 22·5-s − 92·13-s − 104·17-s + 359·25-s + 130·29-s + 396·37-s − 472·41-s − 343·49-s − 518·53-s − 468·61-s − 2.02e3·65-s − 1.09e3·73-s − 2.28e3·85-s − 176·89-s + 594·97-s − 598·101-s + 1.46e3·109-s − 1.32e3·113-s + ⋯ |
| L(s) = 1 | + 1.96·5-s − 1.96·13-s − 1.48·17-s + 2.87·25-s + 0.832·29-s + 1.75·37-s − 1.79·41-s − 49-s − 1.34·53-s − 0.982·61-s − 3.86·65-s − 1.76·73-s − 2.91·85-s − 0.209·89-s + 0.621·97-s − 0.589·101-s + 1.28·109-s − 1.10·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 22 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 92 T + p^{3} T^{2} \) |
| 17 | \( 1 + 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 396 T + p^{3} T^{2} \) |
| 41 | \( 1 + 472 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + 518 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 468 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + 176 T + p^{3} T^{2} \) |
| 97 | \( 1 - 594 T + p^{3} T^{2} \) |
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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
−8.442721614606649735742770949180, −7.32253373224090007499932330360, −6.56858279412549644069088810088, −6.03891654579303787676946986678, −4.94290952356971411996007797812, −4.67721544481997312806899500589, −2.88205342209647591616129487103, −2.33222271120770206960248713590, −1.49537304557110902716672127340, 0,
1.49537304557110902716672127340, 2.33222271120770206960248713590, 2.88205342209647591616129487103, 4.67721544481997312806899500589, 4.94290952356971411996007797812, 6.03891654579303787676946986678, 6.56858279412549644069088810088, 7.32253373224090007499932330360, 8.442721614606649735742770949180
