L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 2·13-s + 14-s + 16-s + 8·17-s + 4·19-s + 4·23-s − 2·26-s − 28-s − 7·29-s + 31-s − 32-s − 8·34-s − 7·37-s − 4·38-s + 9·41-s − 2·43-s − 4·46-s + 7·47-s + 49-s + 2·52-s + 6·53-s + 56-s + 7·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.917·19-s + 0.834·23-s − 0.392·26-s − 0.188·28-s − 1.29·29-s + 0.179·31-s − 0.176·32-s − 1.37·34-s − 1.15·37-s − 0.648·38-s + 1.40·41-s − 0.304·43-s − 0.589·46-s + 1.02·47-s + 1/7·49-s + 0.277·52-s + 0.824·53-s + 0.133·56-s + 0.919·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.640353706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.640353706\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.61940036747590893396786939491, −7.25962308241021337149095326666, −6.44253199593095724676930997534, −5.56513627074496853688010039434, −5.31543366970607218099666451216, −3.93215763349251250331547396654, −3.37893787333662494418110280715, −2.61418171672089279096844744395, −1.45786827768088942485764442992, −0.74411305476223796269134028075,
0.74411305476223796269134028075, 1.45786827768088942485764442992, 2.61418171672089279096844744395, 3.37893787333662494418110280715, 3.93215763349251250331547396654, 5.31543366970607218099666451216, 5.56513627074496853688010039434, 6.44253199593095724676930997534, 7.25962308241021337149095326666, 7.61940036747590893396786939491