| L(s) = 1 | + 3·5-s + 7-s − 11-s + 2·13-s + 2·17-s − 3·19-s − 23-s + 4·25-s − 2·29-s − 5·31-s + 3·35-s + 7·37-s + 7·41-s + 8·43-s + 4·47-s + 49-s − 4·53-s − 3·55-s + 4·59-s + 4·61-s + 6·65-s + 2·67-s + 3·71-s − 77-s + 4·79-s + 2·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.688·19-s − 0.208·23-s + 4/5·25-s − 0.371·29-s − 0.898·31-s + 0.507·35-s + 1.15·37-s + 1.09·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.549·53-s − 0.404·55-s + 0.520·59-s + 0.512·61-s + 0.744·65-s + 0.244·67-s + 0.356·71-s − 0.113·77-s + 0.450·79-s + 0.219·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.880526897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.880526897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.031283237859020875819780729250, −7.44124513817703459179753597530, −6.46686870429689485960980611484, −5.87701928249365151076507487089, −5.42659410911837857479707052937, −4.49441729517652567002436442813, −3.66086872533175093927305837663, −2.52809508545776274555596918001, −1.95304382384510274888764059190, −0.920484220523452143920514394096,
0.920484220523452143920514394096, 1.95304382384510274888764059190, 2.52809508545776274555596918001, 3.66086872533175093927305837663, 4.49441729517652567002436442813, 5.42659410911837857479707052937, 5.87701928249365151076507487089, 6.46686870429689485960980611484, 7.44124513817703459179753597530, 8.031283237859020875819780729250
