| L(s) = 1 | − 5.08·2-s − 7.39·3-s + 17.8·4-s + 8.39·5-s + 37.5·6-s − 14.1·7-s − 49.8·8-s + 27.6·9-s − 42.6·10-s + 11·11-s − 131.·12-s + 72.1·14-s − 62.0·15-s + 110.·16-s − 16.9·17-s − 140.·18-s − 41.1·19-s + 149.·20-s + 104.·21-s − 55.8·22-s + 131.·23-s + 368.·24-s − 54.4·25-s − 4.91·27-s − 252.·28-s − 97.3·29-s + 315.·30-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.42·3-s + 2.22·4-s + 0.751·5-s + 2.55·6-s − 0.766·7-s − 2.20·8-s + 1.02·9-s − 1.34·10-s + 0.301·11-s − 3.16·12-s + 1.37·14-s − 1.06·15-s + 1.73·16-s − 0.241·17-s − 1.84·18-s − 0.496·19-s + 1.67·20-s + 1.09·21-s − 0.541·22-s + 1.19·23-s + 3.13·24-s − 0.435·25-s − 0.0350·27-s − 1.70·28-s − 0.623·29-s + 1.91·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.08T + 8T^{2} \) |
| 3 | \( 1 + 7.39T + 27T^{2} \) |
| 5 | \( 1 - 8.39T + 125T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 17 | \( 1 + 16.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 41.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 97.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 364.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 114.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 507.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 456.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 600.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 665.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 808.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 495.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 494.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 643.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 814.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 701.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 810.T + 9.12e5T^{2} \) |
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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
−8.748172775030618814270784591926, −7.66449201692346270488376920848, −6.87921540440865469641799478147, −6.22746558650578524822475468776, −5.82450406294219151593395692147, −4.57962385961260017544692108890, −2.98287283995700849289217686001, −1.84717712928735204253358688100, −0.869268875748377661242444979845, 0,
0.869268875748377661242444979845, 1.84717712928735204253358688100, 2.98287283995700849289217686001, 4.57962385961260017544692108890, 5.82450406294219151593395692147, 6.22746558650578524822475468776, 6.87921540440865469641799478147, 7.66449201692346270488376920848, 8.748172775030618814270784591926
