| L(s) = 1 | + 1.03·2-s − 3-s − 0.927·4-s − 1.23·5-s − 1.03·6-s + 7-s − 3.03·8-s + 9-s − 1.28·10-s + 3.83·11-s + 0.927·12-s − 5.12·13-s + 1.03·14-s + 1.23·15-s − 1.28·16-s + 8.19·17-s + 1.03·18-s − 5.60·19-s + 1.14·20-s − 21-s + 3.96·22-s − 6.61·23-s + 3.03·24-s − 3.46·25-s − 5.30·26-s − 27-s − 0.927·28-s + ⋯ |
| L(s) = 1 | + 0.732·2-s − 0.577·3-s − 0.463·4-s − 0.554·5-s − 0.422·6-s + 0.377·7-s − 1.07·8-s + 0.333·9-s − 0.405·10-s + 1.15·11-s + 0.267·12-s − 1.42·13-s + 0.276·14-s + 0.319·15-s − 0.320·16-s + 1.98·17-s + 0.244·18-s − 1.28·19-s + 0.257·20-s − 0.218·21-s + 0.845·22-s − 1.38·23-s + 0.618·24-s − 0.692·25-s − 1.04·26-s − 0.192·27-s − 0.175·28-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
| \(L(1)\) |
\(\approx\) |
\(1.337732866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.337732866\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.03T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 8.19T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 + 2.69T + 31T^{2} \) |
| 37 | \( 1 + 4.88T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 - 0.832T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 8.81T + 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 5.94T + 71T^{2} \) |
| 73 | \( 1 + 7.41T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 3.95T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.399492230114222158367406108606, −7.60650784763154573115650202214, −6.96209937179385788809561419907, −5.75389964758494277675490691346, −5.67711489036106053096923146594, −4.37920238436407345688594153092, −4.22678282873772818330576563578, −3.29028813451844400862288623937, −2.01604293946573951915517042928, −0.60635627741595593334033752356,
0.60635627741595593334033752356, 2.01604293946573951915517042928, 3.29028813451844400862288623937, 4.22678282873772818330576563578, 4.37920238436407345688594153092, 5.67711489036106053096923146594, 5.75389964758494277675490691346, 6.96209937179385788809561419907, 7.60650784763154573115650202214, 8.399492230114222158367406108606
