| L(s) = 1 | − 0.713·2-s − 3-s − 1.49·4-s − 2.22·5-s + 0.713·6-s − 7-s + 2.49·8-s + 9-s + 1.59·10-s − 6.12·11-s + 1.49·12-s + 5.59·13-s + 0.713·14-s + 2.22·15-s + 1.20·16-s + 4.66·17-s − 0.713·18-s + 7.62·19-s + 3.32·20-s + 21-s + 4.37·22-s − 6.58·23-s − 2.49·24-s − 0.0321·25-s − 3.99·26-s − 27-s + 1.49·28-s + ⋯ |
| L(s) = 1 | − 0.504·2-s − 0.577·3-s − 0.745·4-s − 0.996·5-s + 0.291·6-s − 0.377·7-s + 0.881·8-s + 0.333·9-s + 0.503·10-s − 1.84·11-s + 0.430·12-s + 1.55·13-s + 0.190·14-s + 0.575·15-s + 0.300·16-s + 1.13·17-s − 0.168·18-s + 1.75·19-s + 0.742·20-s + 0.218·21-s + 0.932·22-s − 1.37·23-s − 0.508·24-s − 0.00643·25-s − 0.783·26-s − 0.192·27-s + 0.281·28-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
| \(L(1)\) |
\(\approx\) |
\(0.3685184324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3685184324\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.713T + 2T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 + 8.11T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 + 3.28T + 41T^{2} \) |
| 43 | \( 1 + 0.975T + 43T^{2} \) |
| 47 | \( 1 + 6.87T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 1.02T + 59T^{2} \) |
| 61 | \( 1 + 7.48T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 + 2.19T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−8.256227479381481439290582990267, −7.69215411649779321176524465389, −7.48767307241949088994812881565, −6.03616434900001503311937374246, −5.47871200961186217736811385133, −4.79339408428596219576940031728, −3.65374438821333622039396175252, −3.35965838845771525661365848811, −1.57305100478258175983036031492, −0.40300528566381358203111077148,
0.40300528566381358203111077148, 1.57305100478258175983036031492, 3.35965838845771525661365848811, 3.65374438821333622039396175252, 4.79339408428596219576940031728, 5.47871200961186217736811385133, 6.03616434900001503311937374246, 7.48767307241949088994812881565, 7.69215411649779321176524465389, 8.256227479381481439290582990267
