| L(s) = 1 | − 2.77·2-s − 3-s + 5.67·4-s + 3.12·5-s + 2.77·6-s + 7-s − 10.1·8-s + 9-s − 8.64·10-s + 6.28·11-s − 5.67·12-s − 4.18·13-s − 2.77·14-s − 3.12·15-s + 16.8·16-s − 2.52·17-s − 2.77·18-s + 7.47·19-s + 17.7·20-s − 21-s − 17.4·22-s + 2.45·23-s + 10.1·24-s + 4.74·25-s + 11.6·26-s − 27-s + 5.67·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 0.577·3-s + 2.83·4-s + 1.39·5-s + 1.13·6-s + 0.377·7-s − 3.60·8-s + 0.333·9-s − 2.73·10-s + 1.89·11-s − 1.63·12-s − 1.16·13-s − 0.740·14-s − 0.805·15-s + 4.22·16-s − 0.613·17-s − 0.653·18-s + 1.71·19-s + 3.96·20-s − 0.218·21-s − 3.71·22-s + 0.512·23-s + 2.08·24-s + 0.948·25-s + 2.27·26-s − 0.192·27-s + 1.07·28-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
| \(L(1)\) |
\(\approx\) |
\(1.003622962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003622962\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.77T + 2T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 11 | \( 1 - 6.28T + 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 - 7.47T + 19T^{2} \) |
| 23 | \( 1 - 2.45T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 0.646T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + 8.93T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 - 0.780T + 53T^{2} \) |
| 59 | \( 1 - 7.64T + 59T^{2} \) |
| 61 | \( 1 - 0.562T + 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 - 9.27T + 89T^{2} \) |
| 97 | \( 1 - 9.03T + 97T^{2} \) |
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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
−8.838403127471205785569082179769, −7.63681587885215242398776252711, −7.09972669678725604985102825090, −6.51527096623615417422567782910, −5.81657410449336474180384779524, −5.08464448762302412747459979189, −3.51387874506719778294169058603, −2.26215127856340854897600304647, −1.66017414076857729873827612979, −0.838951933570368831980395505500,
0.838951933570368831980395505500, 1.66017414076857729873827612979, 2.26215127856340854897600304647, 3.51387874506719778294169058603, 5.08464448762302412747459979189, 5.81657410449336474180384779524, 6.51527096623615417422567782910, 7.09972669678725604985102825090, 7.63681587885215242398776252711, 8.838403127471205785569082179769
