| L(s) = 1 | − 2·3-s + 5-s − 2·7-s + 9-s + 13-s − 2·15-s + 3·17-s − 2·19-s + 4·21-s + 6·23-s − 4·25-s + 4·27-s + 29-s − 10·31-s − 2·35-s − 3·37-s − 2·39-s + 11·41-s − 12·43-s + 45-s − 10·47-s − 3·49-s − 6·51-s + 9·53-s + 4·57-s + 4·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 0.727·17-s − 0.458·19-s + 0.872·21-s + 1.25·23-s − 4/5·25-s + 0.769·27-s + 0.185·29-s − 1.79·31-s − 0.338·35-s − 0.493·37-s − 0.320·39-s + 1.71·41-s − 1.82·43-s + 0.149·45-s − 1.45·47-s − 3/7·49-s − 0.840·51-s + 1.23·53-s + 0.529·57-s + 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9428693061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9428693061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−8.582114976245716932810649330771, −7.58185244895302892767915174290, −6.73790308750564918323910722289, −6.26482973602346525171815142430, −5.48624789496048304682155678642, −5.05518675071975109334736369440, −3.85186303017829101187990588986, −3.06003539288870995137320719209, −1.82293056978851944966844690664, −0.58914386764068834079039361803,
0.58914386764068834079039361803, 1.82293056978851944966844690664, 3.06003539288870995137320719209, 3.85186303017829101187990588986, 5.05518675071975109334736369440, 5.48624789496048304682155678642, 6.26482973602346525171815142430, 6.73790308750564918323910722289, 7.58185244895302892767915174290, 8.582114976245716932810649330771
