| L(s) = 1 | + 0.120·2-s − 0.582·3-s − 1.98·4-s − 1.68·5-s − 0.0700·6-s − 7-s − 0.479·8-s − 2.66·9-s − 0.203·10-s − 0.364·11-s + 1.15·12-s − 0.120·14-s + 0.983·15-s + 3.91·16-s − 3.18·17-s − 0.320·18-s + 1.44·19-s + 3.35·20-s + 0.582·21-s − 0.0438·22-s − 5.08·23-s + 0.279·24-s − 2.15·25-s + 3.29·27-s + 1.98·28-s + 8.19·29-s + 0.118·30-s + ⋯ |
| L(s) = 1 | + 0.0851·2-s − 0.336·3-s − 0.992·4-s − 0.754·5-s − 0.0286·6-s − 0.377·7-s − 0.169·8-s − 0.886·9-s − 0.0642·10-s − 0.109·11-s + 0.333·12-s − 0.0321·14-s + 0.253·15-s + 0.978·16-s − 0.772·17-s − 0.0754·18-s + 0.331·19-s + 0.749·20-s + 0.127·21-s − 0.00935·22-s − 1.05·23-s + 0.0570·24-s − 0.430·25-s + 0.634·27-s + 0.375·28-s + 1.52·29-s + 0.0216·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5806325628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5806325628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.120T + 2T^{2} \) |
| 3 | \( 1 + 0.582T + 3T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 11 | \( 1 + 0.364T + 11T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 + 5.08T + 23T^{2} \) |
| 29 | \( 1 - 8.19T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 + 0.773T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 - 9.36T + 59T^{2} \) |
| 61 | \( 1 + 9.02T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 7.08T + 71T^{2} \) |
| 73 | \( 1 + 2.16T + 73T^{2} \) |
| 79 | \( 1 + 6.88T + 79T^{2} \) |
| 83 | \( 1 - 0.567T + 83T^{2} \) |
| 89 | \( 1 + 1.13T + 89T^{2} \) |
| 97 | \( 1 - 7.92T + 97T^{2} \) |
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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
−9.704459882855299717768213604559, −8.868890207836776417666819405016, −8.249119730907117939455862681677, −7.43001550955776076795907908237, −6.21987727688196312311270458560, −5.54017938097490924817595654151, −4.48231689601386507710951539574, −3.80119307390013222814968547935, −2.65388351025485967126480838234, −0.55495468338389605108456161863,
0.55495468338389605108456161863, 2.65388351025485967126480838234, 3.80119307390013222814968547935, 4.48231689601386507710951539574, 5.54017938097490924817595654151, 6.21987727688196312311270458560, 7.43001550955776076795907908237, 8.249119730907117939455862681677, 8.868890207836776417666819405016, 9.704459882855299717768213604559
