| L(s) = 1 | − 5·5-s + 2·7-s + 34·11-s − 68·13-s + 38·17-s + 4·19-s − 152·23-s + 25·25-s + 46·29-s − 260·31-s − 10·35-s − 312·37-s − 48·41-s − 200·43-s − 104·47-s − 339·49-s + 414·53-s − 170·55-s + 2·59-s − 38·61-s + 340·65-s − 244·67-s − 708·71-s − 378·73-s + 68·77-s − 852·79-s − 844·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.107·7-s + 0.931·11-s − 1.45·13-s + 0.542·17-s + 0.0482·19-s − 1.37·23-s + 1/5·25-s + 0.294·29-s − 1.50·31-s − 0.0482·35-s − 1.38·37-s − 0.182·41-s − 0.709·43-s − 0.322·47-s − 0.988·49-s + 1.07·53-s − 0.416·55-s + 0.00441·59-s − 0.0797·61-s + 0.648·65-s − 0.444·67-s − 1.18·71-s − 0.606·73-s + 0.100·77-s − 1.21·79-s − 1.11·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 46 T + p^{3} T^{2} \) |
| 31 | \( 1 + 260 T + p^{3} T^{2} \) |
| 37 | \( 1 + 312 T + p^{3} T^{2} \) |
| 41 | \( 1 + 48 T + p^{3} T^{2} \) |
| 43 | \( 1 + 200 T + p^{3} T^{2} \) |
| 47 | \( 1 + 104 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 2 T + p^{3} T^{2} \) |
| 61 | \( 1 + 38 T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 378 T + p^{3} T^{2} \) |
| 79 | \( 1 + 852 T + p^{3} T^{2} \) |
| 83 | \( 1 + 844 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1380 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
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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
−10.46731450202322132070595315025, −9.687095346202039244043082137027, −8.692933439812842681244189459040, −7.64479870567527075430293645351, −6.87000851106384288932643139818, −5.59389864145631127028182876766, −4.47024900874516878066698791039, −3.36815809286081251988201683509, −1.80778774766621477588324405382, 0,
1.80778774766621477588324405382, 3.36815809286081251988201683509, 4.47024900874516878066698791039, 5.59389864145631127028182876766, 6.87000851106384288932643139818, 7.64479870567527075430293645351, 8.692933439812842681244189459040, 9.687095346202039244043082137027, 10.46731450202322132070595315025
