| L(s) = 1 | − 3-s − 5-s − 3.62·7-s + 9-s − 6.20·11-s + 0.578·13-s + 15-s + 1.42·17-s − 5.62·19-s + 3.62·21-s − 5.62·23-s + 25-s − 27-s − 2·29-s + 2.57·31-s + 6.20·33-s + 3.62·35-s − 7.83·37-s − 0.578·39-s − 5.25·41-s + 7.25·43-s − 45-s − 6.78·47-s + 6.15·49-s − 1.42·51-s − 2·53-s + 6.20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.37·7-s + 0.333·9-s − 1.87·11-s + 0.160·13-s + 0.258·15-s + 0.344·17-s − 1.29·19-s + 0.791·21-s − 1.17·23-s + 0.200·25-s − 0.192·27-s − 0.371·29-s + 0.463·31-s + 1.08·33-s + 0.613·35-s − 1.28·37-s − 0.0926·39-s − 0.820·41-s + 1.10·43-s − 0.149·45-s − 0.989·47-s + 0.879·49-s − 0.199·51-s − 0.274·53-s + 0.836·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2842024198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2842024198\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + 3.62T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 - 0.578T + 13T^{2} \) |
| 17 | \( 1 - 1.42T + 17T^{2} \) |
| 19 | \( 1 + 5.62T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 2.20T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−8.294973977660165231989832614122, −7.81821216210437628233285155725, −6.90188328829830923063996227670, −6.27125856898126271527822100745, −5.57842873372823564351671767484, −4.78124191459531774452539218241, −3.82336020645274839071274767388, −3.04951313327318384325558064777, −2.07881884058023212436402243347, −0.29432478100593529630591925782,
0.29432478100593529630591925782, 2.07881884058023212436402243347, 3.04951313327318384325558064777, 3.82336020645274839071274767388, 4.78124191459531774452539218241, 5.57842873372823564351671767484, 6.27125856898126271527822100745, 6.90188328829830923063996227670, 7.81821216210437628233285155725, 8.294973977660165231989832614122
