| L(s) = 1 | + 1.55·2-s − 3-s + 0.429·4-s − 1.78·5-s − 1.55·6-s − 7-s − 2.44·8-s + 9-s − 2.77·10-s + 0.515·11-s − 0.429·12-s − 5.03·13-s − 1.55·14-s + 1.78·15-s − 4.67·16-s − 0.0785·17-s + 1.55·18-s + 4.96·19-s − 0.763·20-s + 21-s + 0.803·22-s + 1.02·23-s + 2.44·24-s − 1.83·25-s − 7.84·26-s − 27-s − 0.429·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.214·4-s − 0.796·5-s − 0.636·6-s − 0.377·7-s − 0.865·8-s + 0.333·9-s − 0.877·10-s + 0.155·11-s − 0.123·12-s − 1.39·13-s − 0.416·14-s + 0.459·15-s − 1.16·16-s − 0.0190·17-s + 0.367·18-s + 1.13·19-s − 0.170·20-s + 0.218·21-s + 0.171·22-s + 0.212·23-s + 0.499·24-s − 0.366·25-s − 1.53·26-s − 0.192·27-s − 0.0810·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.429545270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.429545270\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 11 | \( 1 - 0.515T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 17 | \( 1 + 0.0785T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 - 4.62T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 2.98T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 - 7.69T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + 3.98T + 79T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 - 6.33T + 97T^{2} \) |
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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
−8.976939631944392313850939611602, −7.77060638251073536428723979252, −7.26070009396375240314636306556, −6.32888921793338018029315962498, −5.62475840453972102977918368161, −4.79256090248869196216176217446, −4.28918438201091340249971830625, −3.35722657494178865030102560892, −2.53196283820565657023104412921, −0.63213236222520127764979913448,
0.63213236222520127764979913448, 2.53196283820565657023104412921, 3.35722657494178865030102560892, 4.28918438201091340249971830625, 4.79256090248869196216176217446, 5.62475840453972102977918368161, 6.32888921793338018029315962498, 7.26070009396375240314636306556, 7.77060638251073536428723979252, 8.976939631944392313850939611602
