| L(s) = 1 | + 4.34·2-s − 4.30·3-s + 10.9·4-s − 5·5-s − 18.6·6-s + 5.54·7-s + 12.6·8-s − 8.50·9-s − 21.7·10-s + 16.5·11-s − 46.9·12-s − 13·13-s + 24.1·14-s + 21.5·15-s − 32.2·16-s + 103.·17-s − 37.0·18-s − 56.8·19-s − 54.5·20-s − 23.8·21-s + 72.1·22-s + 181.·23-s − 54.4·24-s + 25·25-s − 56.5·26-s + 152.·27-s + 60.4·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.827·3-s + 1.36·4-s − 0.447·5-s − 1.27·6-s + 0.299·7-s + 0.559·8-s − 0.315·9-s − 0.687·10-s + 0.454·11-s − 1.12·12-s − 0.277·13-s + 0.460·14-s + 0.370·15-s − 0.503·16-s + 1.47·17-s − 0.484·18-s − 0.686·19-s − 0.609·20-s − 0.247·21-s + 0.699·22-s + 1.64·23-s − 0.462·24-s + 0.200·25-s − 0.426·26-s + 1.08·27-s + 0.408·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 - 4.34T + 8T^{2} \) |
| 3 | \( 1 + 4.30T + 27T^{2} \) |
| 7 | \( 1 - 5.54T + 343T^{2} \) |
| 11 | \( 1 - 16.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 56.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 127.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 597.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 41.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 66.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 135.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 16.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 669.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 624.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 113.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 657.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 79.9T + 9.12e5T^{2} \) |
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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
−8.246501181335801547374422174760, −7.26611580551935379244436581150, −6.51757050786749171175038226133, −5.80042783424547124444619002554, −5.08874034528654508546848164715, −4.55700201283396493139872074485, −3.50094338974597114302439135903, −2.84666825921099727989975397697, −1.37163441602851763366431761728, 0,
1.37163441602851763366431761728, 2.84666825921099727989975397697, 3.50094338974597114302439135903, 4.55700201283396493139872074485, 5.08874034528654508546848164715, 5.80042783424547124444619002554, 6.51757050786749171175038226133, 7.26611580551935379244436581150, 8.246501181335801547374422174760
