L(s) = 1 | − 2.38·2-s + 3-s + 3.70·4-s + 5-s − 2.38·6-s − 4.07·8-s + 9-s − 2.38·10-s + 3.09·11-s + 3.70·12-s + 5.38·13-s + 15-s + 2.31·16-s + 1.02·17-s − 2.38·18-s − 0.295·19-s + 3.70·20-s − 7.38·22-s − 5.79·23-s − 4.07·24-s − 4·25-s − 12.8·26-s + 27-s + 7.48·29-s − 2.38·30-s − 9.87·31-s + 2.61·32-s + ⋯ |
L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.85·4-s + 0.447·5-s − 0.975·6-s − 1.43·8-s + 0.333·9-s − 0.755·10-s + 0.932·11-s + 1.06·12-s + 1.49·13-s + 0.258·15-s + 0.579·16-s + 0.247·17-s − 0.562·18-s − 0.0677·19-s + 0.828·20-s − 1.57·22-s − 1.20·23-s − 0.831·24-s − 0.800·25-s − 2.52·26-s + 0.192·27-s + 1.38·29-s − 0.436·30-s − 1.77·31-s + 0.461·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.402244838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.402244838\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 - 5.38T + 13T^{2} \) |
| 17 | \( 1 - 1.02T + 17T^{2} \) |
| 19 | \( 1 + 0.295T + 19T^{2} \) |
| 23 | \( 1 + 5.79T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 + 9.87T + 31T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 8.70T + 73T^{2} \) |
| 79 | \( 1 + 3.90T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.397698829736284219733082046841, −7.63367184993799197334868841783, −6.83459985149878170613781171074, −6.29303199960504774829252728219, −5.55155749432516606033743773023, −4.11039615784566442155073413647, −3.52654355486242392301079349896, −2.29444346669550692704697545835, −1.67235401106601636072438354426, −0.824849414721159007893695125748,
0.824849414721159007893695125748, 1.67235401106601636072438354426, 2.29444346669550692704697545835, 3.52654355486242392301079349896, 4.11039615784566442155073413647, 5.55155749432516606033743773023, 6.29303199960504774829252728219, 6.83459985149878170613781171074, 7.63367184993799197334868841783, 8.397698829736284219733082046841