L(s) = 1 | − 3-s + 9-s − 2·11-s + 13-s − 19-s − 2·23-s − 27-s − 8·29-s + 2·33-s − 7·37-s − 39-s + 2·41-s − 4·43-s + 4·53-s + 57-s + 5·61-s + 67-s + 2·69-s − 7·73-s + 79-s + 81-s + 8·83-s + 8·87-s − 12·89-s − 3·97-s − 2·99-s + 101-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.229·19-s − 0.417·23-s − 0.192·27-s − 1.48·29-s + 0.348·33-s − 1.15·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.549·53-s + 0.132·57-s + 0.640·61-s + 0.122·67-s + 0.240·69-s − 0.819·73-s + 0.112·79-s + 1/9·81-s + 0.878·83-s + 0.857·87-s − 1.27·89-s − 0.304·97-s − 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9460093425\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9460093425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−15.06726283819615, −14.85504219736132, −13.99086663881660, −13.52228410275211, −13.02381841384353, −12.48136218019842, −11.99644224399443, −11.31137512130044, −10.97386049324336, −10.32516478840150, −9.906027397890642, −9.216503149223593, −8.610771177053367, −8.009709593958533, −7.403613536590969, −6.847948041525127, −6.228454165834920, −5.480653819434115, −5.266717035792642, −4.326809733721787, −3.808852757238600, −3.033032132119329, −2.149892806107299, −1.486930748737569, −0.3860757444541580,
0.3860757444541580, 1.486930748737569, 2.149892806107299, 3.033032132119329, 3.808852757238600, 4.326809733721787, 5.266717035792642, 5.480653819434115, 6.228454165834920, 6.847948041525127, 7.403613536590969, 8.009709593958533, 8.610771177053367, 9.216503149223593, 9.906027397890642, 10.32516478840150, 10.97386049324336, 11.31137512130044, 11.99644224399443, 12.48136218019842, 13.02381841384353, 13.52228410275211, 13.99086663881660, 14.85504219736132, 15.06726283819615