| L(s) = 1 | + 1.66·2-s + 3·3-s − 5.22·4-s + 1.88·5-s + 4.99·6-s − 7.36·7-s − 22.0·8-s + 9·9-s + 3.13·10-s − 11·11-s − 15.6·12-s + 81.0·13-s − 12.2·14-s + 5.65·15-s + 5.17·16-s − 42.1·17-s + 14.9·18-s + 56.2·19-s − 9.85·20-s − 22.0·21-s − 18.3·22-s − 38.8·23-s − 66.0·24-s − 121.·25-s + 134.·26-s + 27·27-s + 38.4·28-s + ⋯ |
| L(s) = 1 | + 0.588·2-s + 0.577·3-s − 0.653·4-s + 0.168·5-s + 0.339·6-s − 0.397·7-s − 0.973·8-s + 0.333·9-s + 0.0992·10-s − 0.301·11-s − 0.377·12-s + 1.72·13-s − 0.233·14-s + 0.0973·15-s + 0.0809·16-s − 0.601·17-s + 0.196·18-s + 0.679·19-s − 0.110·20-s − 0.229·21-s − 0.177·22-s − 0.352·23-s − 0.561·24-s − 0.971·25-s + 1.01·26-s + 0.192·27-s + 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.778364055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.778364055\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 1.66T + 8T^{2} \) |
| 5 | \( 1 - 1.88T + 125T^{2} \) |
| 7 | \( 1 + 7.36T + 343T^{2} \) |
| 13 | \( 1 - 81.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 56.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 255.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 78.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 332.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 563.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 239.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 634.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 137.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 381.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 492.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 592.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 390.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.70e3T + 9.12e5T^{2} \) |
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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
−8.833080110356945776657790088243, −8.161066728159003429863716066029, −7.27933327872904392714093094965, −6.06155878706198259724777961807, −5.76870771711019923632986960821, −4.55060937782011797098470170241, −3.78881660774196247970442393205, −3.21156068152766354831600854217, −2.00376505185795052243050104210, −0.68209911403999780771433930887,
0.68209911403999780771433930887, 2.00376505185795052243050104210, 3.21156068152766354831600854217, 3.78881660774196247970442393205, 4.55060937782011797098470170241, 5.76870771711019923632986960821, 6.06155878706198259724777961807, 7.27933327872904392714093094965, 8.161066728159003429863716066029, 8.833080110356945776657790088243
