| L(s) = 1 | − 5.15·2-s + 18.5·4-s + 1.27·5-s − 23.0·7-s − 54.4·8-s − 6.55·10-s − 4.21·11-s − 16.3·13-s + 118.·14-s + 132.·16-s − 98.0·17-s − 60.2·19-s + 23.6·20-s + 21.7·22-s + 168.·23-s − 123.·25-s + 84.4·26-s − 428.·28-s + 72.3·29-s − 26.8·31-s − 245.·32-s + 505.·34-s − 29.3·35-s − 86.2·37-s + 310.·38-s − 69.2·40-s − 137.·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.32·4-s + 0.113·5-s − 1.24·7-s − 2.40·8-s − 0.207·10-s − 0.115·11-s − 0.349·13-s + 2.27·14-s + 2.06·16-s − 1.39·17-s − 0.727·19-s + 0.263·20-s + 0.210·22-s + 1.53·23-s − 0.987·25-s + 0.636·26-s − 2.89·28-s + 0.463·29-s − 0.155·31-s − 1.35·32-s + 2.54·34-s − 0.141·35-s − 0.383·37-s + 1.32·38-s − 0.273·40-s − 0.523·41-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
| \(L(2)\) |
\(\approx\) |
\(0.1480442732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1480442732\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.15T + 8T^{2} \) |
| 5 | \( 1 - 1.27T + 125T^{2} \) |
| 7 | \( 1 + 23.0T + 343T^{2} \) |
| 11 | \( 1 + 4.21T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 98.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 60.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 168.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 72.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 26.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 406.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 382.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 173.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 508.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 650.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 291.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 948.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 585.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 587.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 479.T + 9.12e5T^{2} \) |
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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
−8.836730708370061391239477162815, −8.214867753362310699498126582344, −7.17133879139028482123210408661, −6.70140336553037829813870042411, −6.11104998350949900153364774422, −4.77473278125525039459536381954, −3.35806105481309548377135644942, −2.52623012760222212423181312049, −1.59362559106640388929320126760, −0.22396761414646712228997988518,
0.22396761414646712228997988518, 1.59362559106640388929320126760, 2.52623012760222212423181312049, 3.35806105481309548377135644942, 4.77473278125525039459536381954, 6.11104998350949900153364774422, 6.70140336553037829813870042411, 7.17133879139028482123210408661, 8.214867753362310699498126582344, 8.836730708370061391239477162815
