| L(s) = 1 | + 2-s + 4-s − 5-s + 4.89·7-s + 8-s − 10-s − 2.89·13-s + 4.89·14-s + 16-s + 17-s + 4·19-s − 20-s + 4·23-s + 25-s − 2.89·26-s + 4.89·28-s − 6·29-s + 4·31-s + 32-s + 34-s − 4.89·35-s + 6·37-s + 4·38-s − 40-s − 6.89·41-s − 0.898·43-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.85·7-s + 0.353·8-s − 0.316·10-s − 0.804·13-s + 1.30·14-s + 0.250·16-s + 0.242·17-s + 0.917·19-s − 0.223·20-s + 0.834·23-s + 0.200·25-s − 0.568·26-s + 0.925·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.828·35-s + 0.986·37-s + 0.648·38-s − 0.158·40-s − 1.07·41-s − 0.137·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.056907257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.056907257\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4.89T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 6.89T + 41T^{2} \) |
| 43 | \( 1 + 0.898T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 + 0.898T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−9.435965180996115624067997299700, −8.473629864846478120798699660403, −7.58971872121129074728412332697, −7.35156333661789877557729556357, −5.99109365826133501687341959906, −4.97857209698728031677049844408, −4.71593750874726386119638895215, −3.55974972349242469344577540979, −2.40678781580251936782623585700, −1.25796201993737978477703309474,
1.25796201993737978477703309474, 2.40678781580251936782623585700, 3.55974972349242469344577540979, 4.71593750874726386119638895215, 4.97857209698728031677049844408, 5.99109365826133501687341959906, 7.35156333661789877557729556357, 7.58971872121129074728412332697, 8.473629864846478120798699660403, 9.435965180996115624067997299700
