| L(s) = 1 | + 1.55·2-s − 3-s + 0.408·4-s − 3.27·5-s − 1.55·6-s + 2.82·7-s − 2.46·8-s + 9-s − 5.08·10-s − 1.95·11-s − 0.408·12-s − 4.41·13-s + 4.37·14-s + 3.27·15-s − 4.65·16-s + 17-s + 1.55·18-s − 2.97·19-s − 1.33·20-s − 2.82·21-s − 3.03·22-s + 6.58·23-s + 2.46·24-s + 5.72·25-s − 6.84·26-s − 27-s + 1.15·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 0.577·3-s + 0.204·4-s − 1.46·5-s − 0.633·6-s + 1.06·7-s − 0.873·8-s + 0.333·9-s − 1.60·10-s − 0.589·11-s − 0.118·12-s − 1.22·13-s + 1.17·14-s + 0.845·15-s − 1.16·16-s + 0.242·17-s + 0.365·18-s − 0.682·19-s − 0.299·20-s − 0.615·21-s − 0.647·22-s + 1.37·23-s + 0.504·24-s + 1.14·25-s − 1.34·26-s − 0.192·27-s + 0.218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.401482424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401482424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 - 6.58T + 23T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 + 5.15T + 31T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 - 6.77T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 - 7.93T + 73T^{2} \) |
| 79 | \( 1 - 1.39T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 3.61T + 97T^{2} \) |
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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
−8.558347310674620180097555119399, −7.75553931402316934607684170351, −7.28418083474030531551576809920, −6.31519906745011593100171546132, −5.23578656515859831507037966401, −4.80971156747084537228016152856, −4.31680568476276913577144474268, −3.37151979021854658969174366667, −2.38926108975777361393216283430, −0.61227690771153078012147092426,
0.61227690771153078012147092426, 2.38926108975777361393216283430, 3.37151979021854658969174366667, 4.31680568476276913577144474268, 4.80971156747084537228016152856, 5.23578656515859831507037966401, 6.31519906745011593100171546132, 7.28418083474030531551576809920, 7.75553931402316934607684170351, 8.558347310674620180097555119399
