| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 6·13-s + 15-s − 6·17-s − 4·19-s − 21-s − 23-s + 25-s + 27-s − 6·29-s − 35-s − 10·37-s − 6·39-s + 2·41-s + 8·43-s + 45-s − 8·47-s + 49-s − 6·51-s − 10·53-s − 4·57-s + 4·59-s + 14·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.169·35-s − 1.64·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 1.37·53-s − 0.529·57-s + 0.520·59-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224940254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224940254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.69974509363661, −14.40964162808636, −13.85969676037701, −13.05181382934663, −12.96402078264686, −12.39883693904775, −11.72717034627495, −11.06656159142028, −10.54889048687470, −9.979636075372653, −9.417433392452602, −9.142910662530708, −8.443793486134154, −7.884297512062349, −7.201754735703806, −6.706940361896043, −6.308482178109143, −5.278619978035986, −4.987053555323548, −4.128410331218601, −3.667116931190510, −2.659777680774577, −2.277011773131961, −1.734623837779627, −0.3597869941757106,
0.3597869941757106, 1.734623837779627, 2.277011773131961, 2.659777680774577, 3.667116931190510, 4.128410331218601, 4.987053555323548, 5.278619978035986, 6.308482178109143, 6.706940361896043, 7.201754735703806, 7.884297512062349, 8.443793486134154, 9.142910662530708, 9.417433392452602, 9.979636075372653, 10.54889048687470, 11.06656159142028, 11.72717034627495, 12.39883693904775, 12.96402078264686, 13.05181382934663, 13.85969676037701, 14.40964162808636, 14.69974509363661
