L(s) = 1 | + 2.40·3-s − 1.28·5-s − 3.69·7-s + 2.80·9-s − 5.61·11-s + 1.05·13-s − 3.09·15-s + 0.0458·17-s − 19-s − 8.89·21-s + 4.74·23-s − 3.35·25-s − 0.463·27-s − 1.34·29-s + 0.482·31-s − 13.5·33-s + 4.74·35-s − 0.0177·37-s + 2.55·39-s + 9.16·41-s + 10.7·43-s − 3.60·45-s + 6.00·47-s + 6.63·49-s + 0.110·51-s + 0.565·53-s + 7.20·55-s + ⋯ |
L(s) = 1 | + 1.39·3-s − 0.574·5-s − 1.39·7-s + 0.935·9-s − 1.69·11-s + 0.293·13-s − 0.798·15-s + 0.0111·17-s − 0.229·19-s − 1.94·21-s + 0.989·23-s − 0.670·25-s − 0.0891·27-s − 0.250·29-s + 0.0867·31-s − 2.35·33-s + 0.801·35-s − 0.00292·37-s + 0.408·39-s + 1.43·41-s + 1.64·43-s − 0.537·45-s + 0.875·47-s + 0.948·49-s + 0.0154·51-s + 0.0776·53-s + 0.971·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.835478540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.835478540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 - 0.0458T + 17T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 1.34T + 29T^{2} \) |
| 31 | \( 1 - 0.482T + 31T^{2} \) |
| 37 | \( 1 + 0.0177T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 6.00T + 47T^{2} \) |
| 53 | \( 1 - 0.565T + 53T^{2} \) |
| 59 | \( 1 - 4.86T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 83 | \( 1 - 9.71T + 83T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.126979127918133102930019582214, −7.50147042708280324170653049644, −6.95307526032514198398617214108, −5.95212552589147698606737246657, −5.22950558882997491910231748110, −4.01470973822039307489558790119, −3.61094016130681726867102546556, −2.64393567712170095233672393009, −2.43615663442518584221129722498, −0.62541087389065204989902075144,
0.62541087389065204989902075144, 2.43615663442518584221129722498, 2.64393567712170095233672393009, 3.61094016130681726867102546556, 4.01470973822039307489558790119, 5.22950558882997491910231748110, 5.95212552589147698606737246657, 6.95307526032514198398617214108, 7.50147042708280324170653049644, 8.126979127918133102930019582214