| L(s) = 1 | − 3-s − 2·4-s + 9-s − 6·11-s + 2·12-s + 13-s + 4·16-s + 4·17-s + 4·19-s + 4·23-s − 27-s + 6·29-s − 5·31-s + 6·33-s − 2·36-s + 3·37-s − 39-s + 2·41-s − 9·43-s + 12·44-s − 4·48-s − 4·51-s − 2·52-s + 8·53-s − 4·57-s − 2·59-s + 7·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 0.277·13-s + 16-s + 0.970·17-s + 0.917·19-s + 0.834·23-s − 0.192·27-s + 1.11·29-s − 0.898·31-s + 1.04·33-s − 1/3·36-s + 0.493·37-s − 0.160·39-s + 0.312·41-s − 1.37·43-s + 1.80·44-s − 0.577·48-s − 0.560·51-s − 0.277·52-s + 1.09·53-s − 0.529·57-s − 0.260·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.330599417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.330599417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.51920549048757, −13.99011195148401, −13.42782387200462, −13.01336092941510, −12.71018438467237, −12.02073966637213, −11.55067003924981, −10.81705071037732, −10.36623042986189, −9.987667963577194, −9.460455682625593, −8.780316246473586, −8.196300236897871, −7.756066940944654, −7.257922702424131, −6.499130200921298, −5.636148307299615, −5.368492385607178, −4.956037493666237, −4.324166322692799, −3.403198400159025, −3.082645763102306, −2.112860621368240, −1.031127745459295, −0.5312658590995274,
0.5312658590995274, 1.031127745459295, 2.112860621368240, 3.082645763102306, 3.403198400159025, 4.324166322692799, 4.956037493666237, 5.368492385607178, 5.636148307299615, 6.499130200921298, 7.257922702424131, 7.756066940944654, 8.196300236897871, 8.780316246473586, 9.460455682625593, 9.987667963577194, 10.36623042986189, 10.81705071037732, 11.55067003924981, 12.02073966637213, 12.71018438467237, 13.01336092941510, 13.42782387200462, 13.99011195148401, 14.51920549048757
