| L(s) = 1 | − 3.36·3-s + 0.242·5-s + 4.57·7-s + 8.33·9-s − 1.90·11-s − 6.85·13-s − 0.816·15-s − 17-s − 1.47·19-s − 15.3·21-s − 8.75·23-s − 4.94·25-s − 17.9·27-s − 8.58·29-s + 2.36·31-s + 6.42·33-s + 1.10·35-s + 1.79·37-s + 23.0·39-s + 0.885·41-s − 1.60·43-s + 2.02·45-s − 7.99·47-s + 13.8·49-s + 3.36·51-s + 0.915·53-s − 0.463·55-s + ⋯ |
| L(s) = 1 | − 1.94·3-s + 0.108·5-s + 1.72·7-s + 2.77·9-s − 0.575·11-s − 1.90·13-s − 0.210·15-s − 0.242·17-s − 0.338·19-s − 3.35·21-s − 1.82·23-s − 0.988·25-s − 3.45·27-s − 1.59·29-s + 0.425·31-s + 1.11·33-s + 0.187·35-s + 0.294·37-s + 3.69·39-s + 0.138·41-s − 0.244·43-s + 0.301·45-s − 1.16·47-s + 1.98·49-s + 0.471·51-s + 0.125·53-s − 0.0624·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5559252674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5559252674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 3.36T + 3T^{2} \) |
| 5 | \( 1 - 0.242T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 6.85T + 13T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 + 8.75T + 23T^{2} \) |
| 29 | \( 1 + 8.58T + 29T^{2} \) |
| 31 | \( 1 - 2.36T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 0.885T + 41T^{2} \) |
| 43 | \( 1 + 1.60T + 43T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 - 0.915T + 53T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 9.24T + 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 + 9.38T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.67711209880451124784345735236, −7.19356389566658274346178447824, −6.27470564846489146368129565742, −5.55612870276953043561981137426, −5.16093509472174636788010558078, −4.55177252184089491530502495240, −4.02294575125393632557030257149, −2.13104223042060895484586451139, −1.80344610308140508781357735197, −0.39901985556388277029577187209,
0.39901985556388277029577187209, 1.80344610308140508781357735197, 2.13104223042060895484586451139, 4.02294575125393632557030257149, 4.55177252184089491530502495240, 5.16093509472174636788010558078, 5.55612870276953043561981137426, 6.27470564846489146368129565742, 7.19356389566658274346178447824, 7.67711209880451124784345735236
