| L(s) = 1 | + 2-s − 3-s + 4-s − 0.840·5-s − 6-s + 2.87·7-s + 8-s + 9-s − 0.840·10-s + 0.959·11-s − 12-s + 13-s + 2.87·14-s + 0.840·15-s + 16-s − 0.635·17-s + 18-s + 1.12·19-s − 0.840·20-s − 2.87·21-s + 0.959·22-s − 4.87·23-s − 24-s − 4.29·25-s + 26-s − 27-s + 2.87·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.376·5-s − 0.408·6-s + 1.08·7-s + 0.353·8-s + 0.333·9-s − 0.265·10-s + 0.289·11-s − 0.288·12-s + 0.277·13-s + 0.767·14-s + 0.217·15-s + 0.250·16-s − 0.154·17-s + 0.235·18-s + 0.257·19-s − 0.188·20-s − 0.626·21-s + 0.204·22-s − 1.01·23-s − 0.204·24-s − 0.858·25-s + 0.196·26-s − 0.192·27-s + 0.542·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 0.840T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 - 0.959T + 11T^{2} \) |
| 17 | \( 1 + 0.635T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 8.86T + 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 41 | \( 1 + 1.67T + 41T^{2} \) |
| 43 | \( 1 + 9.46T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + 4.83T + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 + 3.53T + 61T^{2} \) |
| 67 | \( 1 - 2.91T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 3.54T + 79T^{2} \) |
| 83 | \( 1 - 4.91T + 83T^{2} \) |
| 89 | \( 1 + 5.72T + 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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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
−7.44828518695306837874522559323, −6.67246553586168454073419145800, −6.01883847479512380851402230556, −5.21689373600989685268928869072, −4.84086015404208504780468752642, −3.87903606230040293508242709576, −3.48475526489911649136442157125, −2.02543963369548393549655203228, −1.53298569203725853134364850259, 0,
1.53298569203725853134364850259, 2.02543963369548393549655203228, 3.48475526489911649136442157125, 3.87903606230040293508242709576, 4.84086015404208504780468752642, 5.21689373600989685268928869072, 6.01883847479512380851402230556, 6.67246553586168454073419145800, 7.44828518695306837874522559323
