| L(s) = 1 | + 2.02·3-s − 4.25·5-s + 0.423·7-s + 1.08·9-s − 4.50·11-s − 4.88·13-s − 8.59·15-s + 5.45·17-s − 4.66·19-s + 0.855·21-s + 2.12·23-s + 13.0·25-s − 3.87·27-s − 3.12·29-s − 9.98·31-s − 9.09·33-s − 1.80·35-s − 9.34·37-s − 9.86·39-s − 2.06·41-s + 11.6·43-s − 4.60·45-s + 2.75·47-s − 6.82·49-s + 11.0·51-s + 5.88·53-s + 19.1·55-s + ⋯ |
| L(s) = 1 | + 1.16·3-s − 1.90·5-s + 0.160·7-s + 0.361·9-s − 1.35·11-s − 1.35·13-s − 2.21·15-s + 1.32·17-s − 1.07·19-s + 0.186·21-s + 0.443·23-s + 2.61·25-s − 0.745·27-s − 0.581·29-s − 1.79·31-s − 1.58·33-s − 0.304·35-s − 1.53·37-s − 1.57·39-s − 0.321·41-s + 1.78·43-s − 0.686·45-s + 0.401·47-s − 0.974·49-s + 1.54·51-s + 0.807·53-s + 2.58·55-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
| \(L(1)\) |
\(\approx\) |
\(0.9533893781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9533893781\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + 4.25T + 5T^{2} \) |
| 7 | \( 1 - 0.423T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 - 2.12T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 9.98T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 + 2.06T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 - 5.88T + 53T^{2} \) |
| 59 | \( 1 + 1.37T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 + 0.712T + 71T^{2} \) |
| 73 | \( 1 - 8.36T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6.13T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 0.654T + 97T^{2} \) |
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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
−7.80267258352960189940540722745, −7.48676283199715039982382272520, −6.92446887079818191124279567304, −5.40682058013823269583571157705, −5.02917841029556000016061475904, −3.96169835055957771220732459278, −3.55651012147072299894182483983, −2.78338442402180660409521603619, −2.09325220677168477603425812304, −0.42452948724107522872534331759,
0.42452948724107522872534331759, 2.09325220677168477603425812304, 2.78338442402180660409521603619, 3.55651012147072299894182483983, 3.96169835055957771220732459278, 5.02917841029556000016061475904, 5.40682058013823269583571157705, 6.92446887079818191124279567304, 7.48676283199715039982382272520, 7.80267258352960189940540722745
