| L(s) = 1 | + 3.25·3-s + 1.44·5-s − 0.736·7-s + 7.60·9-s + 0.631·11-s + 0.659·13-s + 4.69·15-s + 3.60·17-s − 3.07·19-s − 2.39·21-s − 7.24·23-s − 2.91·25-s + 14.9·27-s + 4.33·29-s + 4.66·31-s + 2.05·33-s − 1.06·35-s + 8.87·37-s + 2.14·39-s − 8.77·41-s + 5.16·43-s + 10.9·45-s + 10.0·47-s − 6.45·49-s + 11.7·51-s + 0.0719·53-s + 0.910·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 0.645·5-s − 0.278·7-s + 2.53·9-s + 0.190·11-s + 0.182·13-s + 1.21·15-s + 0.873·17-s − 0.704·19-s − 0.523·21-s − 1.51·23-s − 0.583·25-s + 2.88·27-s + 0.804·29-s + 0.837·31-s + 0.357·33-s − 0.179·35-s + 1.45·37-s + 0.343·39-s − 1.37·41-s + 0.787·43-s + 1.63·45-s + 1.46·47-s − 0.922·49-s + 1.64·51-s + 0.00988·53-s + 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.254166811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.254166811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + 0.736T + 7T^{2} \) |
| 11 | \( 1 - 0.631T + 11T^{2} \) |
| 13 | \( 1 - 0.659T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 - 4.66T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 + 8.77T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 0.0719T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 - 7.19T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
| 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
−8.047806578531162573774519389472, −7.37414684452026818788304748658, −6.46458550071772797262339715746, −5.97548667390493731586928936779, −4.80062853680756682185671411776, −3.99617048771115441799559128108, −3.48844129110673258203552859053, −2.53265325932227721140978687257, −2.10069017618556507484798161539, −1.09637176608263344992757250213,
1.09637176608263344992757250213, 2.10069017618556507484798161539, 2.53265325932227721140978687257, 3.48844129110673258203552859053, 3.99617048771115441799559128108, 4.80062853680756682185671411776, 5.97548667390493731586928936779, 6.46458550071772797262339715746, 7.37414684452026818788304748658, 8.047806578531162573774519389472
