| L(s) = 1 | − 2-s − 3-s + 4-s − 0.745·5-s + 6-s − 1.28·7-s − 8-s + 9-s + 0.745·10-s + 11-s − 12-s + 1.02·13-s + 1.28·14-s + 0.745·15-s + 16-s − 3.94·17-s − 18-s + 1.02·19-s − 0.745·20-s + 1.28·21-s − 22-s + 2.77·23-s + 24-s − 4.44·25-s − 1.02·26-s − 27-s − 1.28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.333·5-s + 0.408·6-s − 0.484·7-s − 0.353·8-s + 0.333·9-s + 0.235·10-s + 0.301·11-s − 0.288·12-s + 0.284·13-s + 0.342·14-s + 0.192·15-s + 0.250·16-s − 0.955·17-s − 0.235·18-s + 0.235·19-s − 0.166·20-s + 0.279·21-s − 0.213·22-s + 0.578·23-s + 0.204·24-s − 0.888·25-s − 0.201·26-s − 0.192·27-s − 0.242·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7331391957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7331391957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 0.745T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 + 3.94T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 - 4.28T + 29T^{2} \) |
| 31 | \( 1 + 5.11T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 - 3.78T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 + 0.306T + 53T^{2} \) |
| 59 | \( 1 + 6.02T + 59T^{2} \) |
| 67 | \( 1 + 9.15T + 67T^{2} \) |
| 71 | \( 1 - 7.77T + 71T^{2} \) |
| 73 | \( 1 + 0.861T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 3.30T + 97T^{2} \) |
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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
−8.543803292583070724401792985779, −7.62518327841286942817111630867, −7.05604498591866515251280052873, −6.31237803412825324186390805199, −5.72418056936473161433523121580, −4.64981004036126353590180222987, −3.84738195692516192801970323602, −2.86768324499019622930677313832, −1.73910424334745096294938686931, −0.56053266485974620319476504499,
0.56053266485974620319476504499, 1.73910424334745096294938686931, 2.86768324499019622930677313832, 3.84738195692516192801970323602, 4.64981004036126353590180222987, 5.72418056936473161433523121580, 6.31237803412825324186390805199, 7.05604498591866515251280052873, 7.62518327841286942817111630867, 8.543803292583070724401792985779
