| L(s) = 1 | + 4·4-s + 9·9-s − 21·11-s − 17·13-s + 16·16-s − 9·17-s + 3·23-s + 25·25-s + 19·31-s + 36·36-s + 39·41-s − 43·43-s − 84·44-s − 78·47-s + 49·49-s − 68·52-s + 63·53-s − 54·59-s + 64·64-s + 91·67-s − 36·68-s − 14·79-s + 81·81-s + 123·83-s + 12·92-s − 193·97-s − 189·99-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s − 1.90·11-s − 1.30·13-s + 16-s − 0.529·17-s + 3/23·23-s + 25-s + 0.612·31-s + 36-s + 0.951·41-s − 43-s − 1.90·44-s − 1.65·47-s + 49-s − 1.30·52-s + 1.18·53-s − 0.915·59-s + 64-s + 1.35·67-s − 0.529·68-s − 0.177·79-s + 81-s + 1.48·83-s + 3/23·92-s − 1.98·97-s − 1.90·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.233644346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233644346\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 21 T + p^{2} T^{2} \) |
| 13 | \( 1 + 17 T + p^{2} T^{2} \) |
| 17 | \( 1 + 9 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 3 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 19 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 39 T + p^{2} T^{2} \) |
| 47 | \( 1 + 78 T + p^{2} T^{2} \) |
| 53 | \( 1 - 63 T + p^{2} T^{2} \) |
| 59 | \( 1 + 54 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 91 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 14 T + p^{2} T^{2} \) |
| 83 | \( 1 - 123 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 193 T + p^{2} T^{2} \) |
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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
−15.68745798247645466816065074550, −14.93830790529910498444437549418, −13.18202833767239569934644068348, −12.29094373488513376318635980589, −10.81120730186677110004926208563, −9.963233779979263944915015630361, −7.888827227322648391797264880663, −6.89295534312803505021028515568, −5.03933818808426973842888393284, −2.54220056186190216196915798797,
2.54220056186190216196915798797, 5.03933818808426973842888393284, 6.89295534312803505021028515568, 7.888827227322648391797264880663, 9.963233779979263944915015630361, 10.81120730186677110004926208563, 12.29094373488513376318635980589, 13.18202833767239569934644068348, 14.93830790529910498444437549418, 15.68745798247645466816065074550
