| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 0.950·5-s + 6·6-s − 15.4·7-s − 8·8-s + 9·9-s − 1.90·10-s + 12.7·11-s − 12·12-s − 86.0·13-s + 30.8·14-s − 2.85·15-s + 16·16-s − 101.·17-s − 18·18-s + 3.80·20-s + 46.2·21-s − 25.5·22-s + 3.47·23-s + 24·24-s − 124.·25-s + 172.·26-s − 27·27-s − 61.6·28-s + 62.2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.0850·5-s + 0.408·6-s − 0.831·7-s − 0.353·8-s + 0.333·9-s − 0.0601·10-s + 0.350·11-s − 0.288·12-s − 1.83·13-s + 0.588·14-s − 0.0490·15-s + 0.250·16-s − 1.44·17-s − 0.235·18-s + 0.0425·20-s + 0.480·21-s − 0.247·22-s + 0.0314·23-s + 0.204·24-s − 0.992·25-s + 1.29·26-s − 0.192·27-s − 0.415·28-s + 0.398·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1112796747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1112796747\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 0.950T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 11 | \( 1 - 12.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 3.47T + 1.21e4T^{2} \) |
| 29 | \( 1 - 62.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 370.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 368.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 100.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 607.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 192.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 157.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 306.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 817.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 93.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 905.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 369.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 5.13T + 7.04e5T^{2} \) |
| 97 | \( 1 - 385.T + 9.12e5T^{2} \) |
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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
−8.968287121229096651472221721787, −7.890793962507096455869479331070, −6.92984203477676407258963362616, −6.73844028141333464721737708937, −5.65369955697836331974316827933, −4.82789419140954632714396388991, −3.77969291845201086849809789064, −2.60203018702164541723556707068, −1.74405665153617932997466719176, −0.16468102941110693987389987301,
0.16468102941110693987389987301, 1.74405665153617932997466719176, 2.60203018702164541723556707068, 3.77969291845201086849809789064, 4.82789419140954632714396388991, 5.65369955697836331974316827933, 6.73844028141333464721737708937, 6.92984203477676407258963362616, 7.890793962507096455869479331070, 8.968287121229096651472221721787
