| L(s) = 1 | − 2.70·2-s + 5.32·4-s + 0.470·7-s − 8.99·8-s + 3.18·11-s + 0.563·13-s − 1.27·14-s + 13.7·16-s + 1.70·17-s + 3.74·19-s − 8.61·22-s + 2.26·23-s − 1.52·26-s + 2.50·28-s + 8.32·29-s + 5.43·31-s − 19.0·32-s − 4.61·34-s + 1.02·37-s − 10.1·38-s − 1.47·41-s + 6.72·43-s + 16.9·44-s − 6.14·46-s + 4.43·47-s − 6.77·49-s + 3·52-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 2.66·4-s + 0.177·7-s − 3.18·8-s + 0.959·11-s + 0.156·13-s − 0.340·14-s + 3.42·16-s + 0.413·17-s + 0.858·19-s − 1.83·22-s + 0.473·23-s − 0.299·26-s + 0.473·28-s + 1.54·29-s + 0.976·31-s − 3.37·32-s − 0.791·34-s + 0.167·37-s − 1.64·38-s − 0.229·41-s + 1.02·43-s + 2.55·44-s − 0.905·46-s + 0.646·47-s − 0.968·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045752633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045752633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 7 | \( 1 - 0.470T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 0.563T + 13T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 1.02T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 0.126T + 61T^{2} \) |
| 67 | \( 1 + 2.79T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 9.78T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 6.20T + 89T^{2} \) |
| 97 | \( 1 - 8.45T + 97T^{2} \) |
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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
−8.369955380743841393367246115278, −7.51710411973927960187807377156, −7.02418802994809150439481167846, −6.31966917142960509767566014203, −5.63700419715077636150024685890, −4.42681421437838913791884669373, −3.25139287792593432545764781594, −2.53744037953481274395135150771, −1.38137514730308622181104534504, −0.828489530382110147773615584260,
0.828489530382110147773615584260, 1.38137514730308622181104534504, 2.53744037953481274395135150771, 3.25139287792593432545764781594, 4.42681421437838913791884669373, 5.63700419715077636150024685890, 6.31966917142960509767566014203, 7.02418802994809150439481167846, 7.51710411973927960187807377156, 8.369955380743841393367246115278
