L(s) = 1 | + 6·11-s + 6·17-s − 2·19-s − 5·25-s − 6·41-s + 10·43-s − 7·49-s + 6·59-s + 14·67-s − 2·73-s + 18·83-s + 18·89-s + 10·97-s + 6·107-s − 18·113-s + ⋯ |
L(s) = 1 | + 1.80·11-s + 1.45·17-s − 0.458·19-s − 25-s − 0.937·41-s + 1.52·43-s − 49-s + 0.781·59-s + 1.71·67-s − 0.234·73-s + 1.97·83-s + 1.90·89-s + 1.01·97-s + 0.580·107-s − 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.056695321\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.056695321\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.111811254215445438085933089837, −8.227029907063207442212793556286, −7.49800309626174308942277426392, −6.58492595927970716736421063363, −6.00725685310885624467606512287, −5.04834818214897768625909783776, −3.97459158177538150872708928083, −3.44733369104578574594522067100, −2.04286410288724293558493803436, −0.986409188428603835741707173143,
0.986409188428603835741707173143, 2.04286410288724293558493803436, 3.44733369104578574594522067100, 3.97459158177538150872708928083, 5.04834818214897768625909783776, 6.00725685310885624467606512287, 6.58492595927970716736421063363, 7.49800309626174308942277426392, 8.227029907063207442212793556286, 9.111811254215445438085933089837