| L(s) = 1 | + 3-s + 1.78·5-s + 4.79·7-s + 9-s + 5.70·11-s − 3.70·13-s + 1.78·15-s − 5.05·17-s − 2.12·19-s + 4.79·21-s − 8.04·23-s − 1.79·25-s + 27-s − 6.25·29-s + 5.99·31-s + 5.70·33-s + 8.58·35-s + 5.45·37-s − 3.70·39-s + 8.42·41-s − 10.0·43-s + 1.78·45-s − 6.36·47-s + 16.0·49-s − 5.05·51-s − 7.49·53-s + 10.2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.800·5-s + 1.81·7-s + 0.333·9-s + 1.72·11-s − 1.02·13-s + 0.462·15-s − 1.22·17-s − 0.486·19-s + 1.04·21-s − 1.67·23-s − 0.359·25-s + 0.192·27-s − 1.16·29-s + 1.07·31-s + 0.993·33-s + 1.45·35-s + 0.896·37-s − 0.593·39-s + 1.31·41-s − 1.53·43-s + 0.266·45-s − 0.928·47-s + 2.28·49-s − 0.707·51-s − 1.03·53-s + 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.552778801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.552778801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 4.79T + 7T^{2} \) |
| 11 | \( 1 - 5.70T + 11T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 + 8.04T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 - 5.45T + 37T^{2} \) |
| 41 | \( 1 - 8.42T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 + 7.49T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 71 | \( 1 + 0.291T + 71T^{2} \) |
| 73 | \( 1 - 3.32T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 + 8.63T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
| 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
−10.01873056307647019465459364155, −9.436312831391410080351769317687, −8.531298940282317670756224489487, −7.88342772990869158373152273876, −6.81934565972445704696010908929, −5.89738352806542964796329360249, −4.65026141665517062976260240413, −4.08309539823270834798204444146, −2.22390540113958314242010144471, −1.66884682319572360530551967629,
1.66884682319572360530551967629, 2.22390540113958314242010144471, 4.08309539823270834798204444146, 4.65026141665517062976260240413, 5.89738352806542964796329360249, 6.81934565972445704696010908929, 7.88342772990869158373152273876, 8.531298940282317670756224489487, 9.436312831391410080351769317687, 10.01873056307647019465459364155
