| L(s) = 1 | + 4.20·2-s + 9.70·4-s − 14.3·5-s − 14.1·7-s + 7.18·8-s − 60.2·10-s − 40.6·11-s − 64.6·13-s − 59.5·14-s − 47.4·16-s − 38.1·17-s − 42.2·19-s − 138.·20-s − 171.·22-s + 189.·23-s + 79.9·25-s − 271.·26-s − 137.·28-s + 50.4·29-s + 242.·31-s − 257.·32-s − 160.·34-s + 202.·35-s + 339.·37-s − 177.·38-s − 102.·40-s − 210.·41-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 1.21·4-s − 1.28·5-s − 0.764·7-s + 0.317·8-s − 1.90·10-s − 1.11·11-s − 1.37·13-s − 1.13·14-s − 0.740·16-s − 0.544·17-s − 0.509·19-s − 1.55·20-s − 1.65·22-s + 1.71·23-s + 0.639·25-s − 2.05·26-s − 0.927·28-s + 0.322·29-s + 1.40·31-s − 1.41·32-s − 0.809·34-s + 0.979·35-s + 1.50·37-s − 0.758·38-s − 0.406·40-s − 0.800·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.567406923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567406923\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 4.20T + 8T^{2} \) |
| 5 | \( 1 + 14.3T + 125T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 11 | \( 1 + 40.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 38.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 50.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 133.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 264.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 437.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 610.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 252.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 50.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 426.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 568.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 883.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 947.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 916.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3T + 9.12e5T^{2} \) |
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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
−8.546778166472879675043337754052, −7.71514735092815785257278176191, −6.96482575097469749308990639251, −6.36681500324139470976504895947, −5.16236073201542215077365024209, −4.72696440389829837741527983096, −3.94802660267808666700634187112, −2.93851634597104444996484118995, −2.57024791763983177438050128755, −0.41829625443227922901784017741,
0.41829625443227922901784017741, 2.57024791763983177438050128755, 2.93851634597104444996484118995, 3.94802660267808666700634187112, 4.72696440389829837741527983096, 5.16236073201542215077365024209, 6.36681500324139470976504895947, 6.96482575097469749308990639251, 7.71514735092815785257278176191, 8.546778166472879675043337754052
