L(s) = 1 | + 2-s + 4-s + 3.64·5-s + 8-s + 3.64·10-s − 11-s − 5·13-s + 16-s + 6·17-s + 0.354·19-s + 3.64·20-s − 22-s + 3.64·23-s + 8.29·25-s − 5·26-s + 4.29·29-s + 4·31-s + 32-s + 6·34-s − 1.64·37-s + 0.354·38-s + 3.64·40-s − 4.93·41-s − 4·43-s − 44-s + 3.64·46-s + 13.2·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.63·5-s + 0.353·8-s + 1.15·10-s − 0.301·11-s − 1.38·13-s + 0.250·16-s + 1.45·17-s + 0.0812·19-s + 0.815·20-s − 0.213·22-s + 0.760·23-s + 1.65·25-s − 0.980·26-s + 0.796·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.270·37-s + 0.0574·38-s + 0.576·40-s − 0.771·41-s − 0.609·43-s − 0.150·44-s + 0.537·46-s + 1.93·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.970540323\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.970540323\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 3.64T + 5T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 0.354T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 3.64T + 53T^{2} \) |
| 59 | \( 1 + 0.645T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 + 9.64T + 71T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 5.70T + 97T^{2} \) |
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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
−7.42936605814683152705648170066, −6.90953797554558321376812518546, −6.15187726486640824680597671163, −5.48258864815507642787907927932, −5.13968927244974983056683892428, −4.41487810670351610105115296256, −3.17874961556046510490202253988, −2.68823030177219340408620936685, −1.92947958163796855151529363336, −0.986360983275491990916946305414,
0.986360983275491990916946305414, 1.92947958163796855151529363336, 2.68823030177219340408620936685, 3.17874961556046510490202253988, 4.41487810670351610105115296256, 5.13968927244974983056683892428, 5.48258864815507642787907927932, 6.15187726486640824680597671163, 6.90953797554558321376812518546, 7.42936605814683152705648170066