| L(s) = 1 | − 1.12·3-s − 1.74·9-s − 4.94·11-s + 4.64·13-s − 5.72·17-s − 7.53·19-s + 6.87·23-s + 5.31·27-s + 8.11·29-s + 3.87·31-s + 5.54·33-s − 5.18·37-s − 5.20·39-s + 9.45·41-s + 0.706·43-s + 2.20·47-s + 6.42·51-s + 1.32·53-s + 8.45·57-s + 3.94·59-s + 6.07·61-s + 8.24·67-s − 7.71·69-s − 4.50·71-s + 2.93·73-s − 11.0·79-s − 0.746·81-s + ⋯ |
| L(s) = 1 | − 0.647·3-s − 0.580·9-s − 1.49·11-s + 1.28·13-s − 1.38·17-s − 1.72·19-s + 1.43·23-s + 1.02·27-s + 1.50·29-s + 0.696·31-s + 0.965·33-s − 0.852·37-s − 0.833·39-s + 1.47·41-s + 0.107·43-s + 0.321·47-s + 0.899·51-s + 0.181·53-s + 1.12·57-s + 0.513·59-s + 0.777·61-s + 1.00·67-s − 0.928·69-s − 0.535·71-s + 0.343·73-s − 1.24·79-s − 0.0829·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.12T + 3T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 19 | \( 1 + 7.53T + 19T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 - 8.11T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 - 9.45T + 41T^{2} \) |
| 43 | \( 1 - 0.706T + 43T^{2} \) |
| 47 | \( 1 - 2.20T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 - 6.07T + 61T^{2} \) |
| 67 | \( 1 - 8.24T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 - 2.93T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 5.27T + 83T^{2} \) |
| 89 | \( 1 - 9.49T + 89T^{2} \) |
| 97 | \( 1 + 1.74T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.16065949091256982199266542688, −6.48601271046069964880307395509, −6.08580817029679755897426616938, −5.24668026964665208716679583612, −4.70093604915328285837122140591, −3.94626490563244064818293748379, −2.79002849218247765243940761822, −2.38532749466397201652698587488, −0.987728202579633771055884004360, 0,
0.987728202579633771055884004360, 2.38532749466397201652698587488, 2.79002849218247765243940761822, 3.94626490563244064818293748379, 4.70093604915328285837122140591, 5.24668026964665208716679583612, 6.08580817029679755897426616938, 6.48601271046069964880307395509, 7.16065949091256982199266542688
