| L(s) = 1 | − 2.59·2-s + 0.175·3-s + 4.72·4-s + 3.23·5-s − 0.456·6-s + 2.53·7-s − 7.05·8-s − 2.96·9-s − 8.38·10-s + 5.30·11-s + 0.830·12-s − 5.29·13-s − 6.57·14-s + 0.568·15-s + 8.85·16-s + 2.29·17-s + 7.69·18-s − 3.46·19-s + 15.2·20-s + 0.446·21-s − 13.7·22-s + 23-s − 1.24·24-s + 5.45·25-s + 13.7·26-s − 1.05·27-s + 11.9·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.101·3-s + 2.36·4-s + 1.44·5-s − 0.186·6-s + 0.958·7-s − 2.49·8-s − 0.989·9-s − 2.65·10-s + 1.60·11-s + 0.239·12-s − 1.46·13-s − 1.75·14-s + 0.146·15-s + 2.21·16-s + 0.557·17-s + 1.81·18-s − 0.795·19-s + 3.41·20-s + 0.0973·21-s − 2.93·22-s + 0.208·23-s − 0.253·24-s + 1.09·25-s + 2.69·26-s − 0.202·27-s + 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321693244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321693244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 349 | \( 1 + T \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 - 0.175T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 29 | \( 1 + 8.54T + 29T^{2} \) |
| 31 | \( 1 - 5.22T + 31T^{2} \) |
| 37 | \( 1 - 9.38T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 3.98T + 43T^{2} \) |
| 47 | \( 1 + 2.42T + 47T^{2} \) |
| 53 | \( 1 - 7.85T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 7.45T + 67T^{2} \) |
| 71 | \( 1 + 3.92T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 1.45T + 79T^{2} \) |
| 83 | \( 1 - 6.69T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.85519733227219597803380703574, −7.48648576465257044718129313532, −6.48313405328057924961012956753, −6.10515756180375877273251468071, −5.35415443531411628645365380268, −4.33492920186579334904546512786, −2.91753906987768947797846916625, −2.20382566168604748403107723448, −1.69820847086196300491158097719, −0.76183897768778390944607962152,
0.76183897768778390944607962152, 1.69820847086196300491158097719, 2.20382566168604748403107723448, 2.91753906987768947797846916625, 4.33492920186579334904546512786, 5.35415443531411628645365380268, 6.10515756180375877273251468071, 6.48313405328057924961012956753, 7.48648576465257044718129313532, 7.85519733227219597803380703574
