| L(s) = 1 | + 4·7-s − 6·11-s − 2·13-s − 6·17-s − 6·19-s − 23-s − 5·25-s + 6·29-s − 8·37-s − 6·41-s − 2·43-s + 8·47-s + 9·49-s + 8·53-s − 4·59-s − 4·61-s + 2·67-s + 8·71-s + 6·73-s − 24·77-s + 12·79-s − 10·83-s − 10·89-s − 8·91-s − 18·97-s − 6·101-s + 8·103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.80·11-s − 0.554·13-s − 1.45·17-s − 1.37·19-s − 0.208·23-s − 25-s + 1.11·29-s − 1.31·37-s − 0.937·41-s − 0.304·43-s + 1.16·47-s + 9/7·49-s + 1.09·53-s − 0.520·59-s − 0.512·61-s + 0.244·67-s + 0.949·71-s + 0.702·73-s − 2.73·77-s + 1.35·79-s − 1.09·83-s − 1.05·89-s − 0.838·91-s − 1.82·97-s − 0.597·101-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.601348166683304914056585059265, −8.320236055459786854807713491348, −7.51799242171937345337021049954, −6.64787884445607472888119813091, −5.46759063803448033131660005497, −4.86456749161076388512551590669, −4.14218283371488576671439383744, −2.53210728436057851679498975053, −1.94477350504889909491080153121, 0,
1.94477350504889909491080153121, 2.53210728436057851679498975053, 4.14218283371488576671439383744, 4.86456749161076388512551590669, 5.46759063803448033131660005497, 6.64787884445607472888119813091, 7.51799242171937345337021049954, 8.320236055459786854807713491348, 8.601348166683304914056585059265
