| L(s) = 1 | + 3·3-s − 5-s + 6·9-s + 2·11-s − 3·15-s + 4·17-s − 6·19-s + 3·23-s + 25-s + 9·27-s − 9·29-s + 4·31-s + 6·33-s + 4·37-s + 7·41-s + 5·43-s − 6·45-s − 8·47-s + 12·51-s + 2·53-s − 2·55-s − 18·57-s + 10·59-s + 61-s + 9·67-s + 9·69-s + 2·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s + 0.603·11-s − 0.774·15-s + 0.970·17-s − 1.37·19-s + 0.625·23-s + 1/5·25-s + 1.73·27-s − 1.67·29-s + 0.718·31-s + 1.04·33-s + 0.657·37-s + 1.09·41-s + 0.762·43-s − 0.894·45-s − 1.16·47-s + 1.68·51-s + 0.274·53-s − 0.269·55-s − 2.38·57-s + 1.30·59-s + 0.128·61-s + 1.09·67-s + 1.08·69-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.484466987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.484466987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.84483950066143, −15.13705839138768, −14.84647136549764, −14.47793826673126, −13.92380790726831, −13.21133999001461, −12.78318869160244, −12.34956986274806, −11.40368411078181, −10.96760772605436, −10.10136266572857, −9.557133539625226, −9.136552155812015, −8.433400121093676, −8.082535198255559, −7.457027639681002, −6.907517315649189, −6.156733740387075, −5.231315739242097, −4.246698187066531, −3.921876339445850, −3.242752073394177, −2.510975381782087, −1.831827280274288, −0.8540635080166242,
0.8540635080166242, 1.831827280274288, 2.510975381782087, 3.242752073394177, 3.921876339445850, 4.246698187066531, 5.231315739242097, 6.156733740387075, 6.907517315649189, 7.457027639681002, 8.082535198255559, 8.433400121093676, 9.136552155812015, 9.557133539625226, 10.10136266572857, 10.96760772605436, 11.40368411078181, 12.34956986274806, 12.78318869160244, 13.21133999001461, 13.92380790726831, 14.47793826673126, 14.84647136549764, 15.13705839138768, 15.84483950066143
