| L(s) = 1 | + 2·2-s + 2·4-s − 3·5-s − 6·10-s + 5·11-s − 2·13-s − 4·16-s − 6·17-s + 6·19-s − 6·20-s + 10·22-s − 9·23-s + 4·25-s − 4·26-s − 29-s + 2·31-s − 8·32-s − 12·34-s + 37-s + 12·38-s + 6·41-s + 2·43-s + 10·44-s − 18·46-s + 47-s + 8·50-s − 4·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.34·5-s − 1.89·10-s + 1.50·11-s − 0.554·13-s − 16-s − 1.45·17-s + 1.37·19-s − 1.34·20-s + 2.13·22-s − 1.87·23-s + 4/5·25-s − 0.784·26-s − 0.185·29-s + 0.359·31-s − 1.41·32-s − 2.05·34-s + 0.164·37-s + 1.94·38-s + 0.937·41-s + 0.304·43-s + 1.50·44-s − 2.65·46-s + 0.145·47-s + 1.13·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20727 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20727 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.598105714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.598105714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.53918820203634, −15.06400875393024, −14.30732195580735, −14.13064832123647, −13.59395171908113, −12.77273308007040, −12.30695175422308, −11.87866005460483, −11.43394562409260, −11.19417449234352, −10.12488532688118, −9.411311219836978, −8.926324696792052, −8.208681649276775, −7.422923375953189, −7.095944583457852, −6.205819935876982, −5.950186267532285, −4.853747522084696, −4.435240647364345, −3.934792046170743, −3.491379695702104, −2.663517723154641, −1.796971006382188, −0.5116401416806858,
0.5116401416806858, 1.796971006382188, 2.663517723154641, 3.491379695702104, 3.934792046170743, 4.435240647364345, 4.853747522084696, 5.950186267532285, 6.205819935876982, 7.095944583457852, 7.422923375953189, 8.208681649276775, 8.926324696792052, 9.411311219836978, 10.12488532688118, 11.19417449234352, 11.43394562409260, 11.87866005460483, 12.30695175422308, 12.77273308007040, 13.59395171908113, 14.13064832123647, 14.30732195580735, 15.06400875393024, 15.53918820203634
