| L(s) = 1 | + 1.41·3-s + 1.41·7-s − 0.999·9-s + 2.82·11-s − 2·13-s − 2·17-s − 5.65·19-s + 2.00·21-s + 1.41·23-s − 5.65·27-s + 8·29-s − 8.48·31-s + 4.00·33-s − 10·37-s − 2.82·39-s − 7.07·43-s − 9.89·47-s − 5·49-s − 2.82·51-s − 2·53-s − 8.00·57-s − 5.65·59-s + 10·61-s − 1.41·63-s + 1.41·67-s + 2.00·69-s + 14.1·71-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.534·7-s − 0.333·9-s + 0.852·11-s − 0.554·13-s − 0.485·17-s − 1.29·19-s + 0.436·21-s + 0.294·23-s − 1.08·27-s + 1.48·29-s − 1.52·31-s + 0.696·33-s − 1.64·37-s − 0.452·39-s − 1.07·43-s − 1.44·47-s − 0.714·49-s − 0.396·51-s − 0.274·53-s − 1.05·57-s − 0.736·59-s + 1.28·61-s − 0.178·63-s + 0.172·67-s + 0.240·69-s + 1.67·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
| 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
−7.924856596453866057467642175425, −6.81395972480546299041132073171, −6.57806798219420119004523404524, −5.39600081196474816560175108297, −4.76923727563996389514491124801, −3.90021435992391830790206852967, −3.21411533513852415825541645811, −2.24467587547750002395099680213, −1.61066136719801239993793103110, 0,
1.61066136719801239993793103110, 2.24467587547750002395099680213, 3.21411533513852415825541645811, 3.90021435992391830790206852967, 4.76923727563996389514491124801, 5.39600081196474816560175108297, 6.57806798219420119004523404524, 6.81395972480546299041132073171, 7.924856596453866057467642175425
