L(s) = 1 | + 3-s − 3·7-s + 9-s − 2·11-s − 3·13-s − 6·17-s + 7·19-s − 3·21-s − 6·23-s + 27-s + 2·29-s − 5·31-s − 2·33-s + 10·37-s − 3·39-s + 12·41-s + 3·43-s + 10·47-s + 2·49-s − 6·51-s + 7·57-s + 6·59-s + 13·61-s − 3·63-s + 7·67-s − 6·69-s − 4·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 0.832·13-s − 1.45·17-s + 1.60·19-s − 0.654·21-s − 1.25·23-s + 0.192·27-s + 0.371·29-s − 0.898·31-s − 0.348·33-s + 1.64·37-s − 0.480·39-s + 1.87·41-s + 0.457·43-s + 1.45·47-s + 2/7·49-s − 0.840·51-s + 0.927·57-s + 0.781·59-s + 1.66·61-s − 0.377·63-s + 0.855·67-s − 0.722·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.606426312\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.606426312\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.207378328669693576016327313651, −7.52102134607961268724297202577, −7.00676888326165244092177400841, −6.11735811920038969417895209815, −5.42894379872931774283445185861, −4.40014462482591421733437837194, −3.73565672996937699736714825324, −2.68955062076436020428485026343, −2.30823427067005295301224493376, −0.64608893255567163621486832127,
0.64608893255567163621486832127, 2.30823427067005295301224493376, 2.68955062076436020428485026343, 3.73565672996937699736714825324, 4.40014462482591421733437837194, 5.42894379872931774283445185861, 6.11735811920038969417895209815, 7.00676888326165244092177400841, 7.52102134607961268724297202577, 8.207378328669693576016327313651