| L(s) = 1 | − 0.0484·2-s − 3-s − 1.99·4-s + 1.78·5-s + 0.0484·6-s + 7-s + 0.193·8-s + 9-s − 0.0866·10-s + 3.36·11-s + 1.99·12-s − 2.05·13-s − 0.0484·14-s − 1.78·15-s + 3.98·16-s − 1.59·17-s − 0.0484·18-s + 5.14·19-s − 3.57·20-s − 21-s − 0.162·22-s + 6.94·23-s − 0.193·24-s − 1.79·25-s + 0.0994·26-s − 27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | − 0.0342·2-s − 0.577·3-s − 0.998·4-s + 0.800·5-s + 0.0197·6-s + 0.377·7-s + 0.0684·8-s + 0.333·9-s − 0.0274·10-s + 1.01·11-s + 0.576·12-s − 0.569·13-s − 0.0129·14-s − 0.461·15-s + 0.996·16-s − 0.386·17-s − 0.0114·18-s + 1.17·19-s − 0.799·20-s − 0.218·21-s − 0.0347·22-s + 1.44·23-s − 0.0395·24-s − 0.359·25-s + 0.0195·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578202124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578202124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0484T + 2T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 + 1.59T + 17T^{2} \) |
| 19 | \( 1 - 5.14T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 - 4.21T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 4.67T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 3.69T + 83T^{2} \) |
| 89 | \( 1 - 1.98T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−8.725710189707868386299872980504, −7.60827763633698381252257681624, −7.00592811811199961479844689913, −6.05960563877259739391452752368, −5.39853793759637048332429632397, −4.81295031653036330724615054507, −4.05869202098266765531532863382, −3.02240296096428906862869525783, −1.68913358103341701237074974338, −0.804498669664489498586182876123,
0.804498669664489498586182876123, 1.68913358103341701237074974338, 3.02240296096428906862869525783, 4.05869202098266765531532863382, 4.81295031653036330724615054507, 5.39853793759637048332429632397, 6.05960563877259739391452752368, 7.00592811811199961479844689913, 7.60827763633698381252257681624, 8.725710189707868386299872980504
