L(s) = 1 | − 7·7-s + 54·11-s − 55·13-s − 18·17-s − 25·19-s + 18·23-s + 54·29-s − 271·31-s + 314·37-s + 360·41-s − 163·43-s − 522·47-s − 294·49-s + 36·53-s − 126·59-s + 47·61-s − 343·67-s + 1.08e3·71-s − 1.05e3·73-s − 378·77-s − 568·79-s − 1.42e3·83-s − 1.44e3·89-s + 385·91-s − 439·97-s − 828·101-s + 548·103-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.48·11-s − 1.17·13-s − 0.256·17-s − 0.301·19-s + 0.163·23-s + 0.345·29-s − 1.57·31-s + 1.39·37-s + 1.37·41-s − 0.578·43-s − 1.62·47-s − 6/7·49-s + 0.0933·53-s − 0.278·59-s + 0.0986·61-s − 0.625·67-s + 1.80·71-s − 1.68·73-s − 0.559·77-s − 0.808·79-s − 1.88·83-s − 1.71·89-s + 0.443·91-s − 0.459·97-s − 0.815·101-s + 0.524·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + p T + p^{3} T^{2} \) |
| 11 | \( 1 - 54 T + p^{3} T^{2} \) |
| 13 | \( 1 + 55 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 25 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 271 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 + 163 T + p^{3} T^{2} \) |
| 47 | \( 1 + 522 T + p^{3} T^{2} \) |
| 53 | \( 1 - 36 T + p^{3} T^{2} \) |
| 59 | \( 1 + 126 T + p^{3} T^{2} \) |
| 61 | \( 1 - 47 T + p^{3} T^{2} \) |
| 67 | \( 1 + 343 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1080 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1054 T + p^{3} T^{2} \) |
| 79 | \( 1 + 568 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1422 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1440 T + p^{3} T^{2} \) |
| 97 | \( 1 + 439 T + p^{3} T^{2} \) |
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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
−9.443470050684727502275732320038, −8.584326934552546357656039390523, −7.48331874230766809047106608277, −6.74259937258134428263388020746, −5.93865693937674717490042953173, −4.74951131124620748484117996292, −3.89543702248029544028614716836, −2.74183560392729761193269454243, −1.48237133102346867222244711648, 0,
1.48237133102346867222244711648, 2.74183560392729761193269454243, 3.89543702248029544028614716836, 4.74951131124620748484117996292, 5.93865693937674717490042953173, 6.74259937258134428263388020746, 7.48331874230766809047106608277, 8.584326934552546357656039390523, 9.443470050684727502275732320038