L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 5.87·11-s − 12-s + 5.36·13-s + 14-s − 15-s + 16-s − 0.508·17-s + 18-s + 1.49·19-s + 20-s − 21-s + 5.87·22-s − 23-s − 24-s + 25-s + 5.36·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.77·11-s − 0.288·12-s + 1.48·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.123·17-s + 0.235·18-s + 0.342·19-s + 0.223·20-s − 0.218·21-s + 1.25·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s + 1.05·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.767614113\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.767614113\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 5.87T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 + 0.508T + 17T^{2} \) |
| 19 | \( 1 - 1.49T + 19T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 + 0.983T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 5.87T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.21T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 7.39T + 97T^{2} \) |
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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
−8.293320525807857831327601522864, −7.25311558767002929763351719735, −6.62397861365678179187300773994, −5.96278981400793969504747030319, −5.57738678628620497288918642812, −4.38580496571612401553306097828, −4.02652014403058145512949721579, −3.05973110777800318461065969525, −1.71514378027106921447602565493, −1.13810353540485618513726464980,
1.13810353540485618513726464980, 1.71514378027106921447602565493, 3.05973110777800318461065969525, 4.02652014403058145512949721579, 4.38580496571612401553306097828, 5.57738678628620497288918642812, 5.96278981400793969504747030319, 6.62397861365678179187300773994, 7.25311558767002929763351719735, 8.293320525807857831327601522864