| L(s) = 1 | − 2-s − 3.32·3-s + 4-s + 4.15·5-s + 3.32·6-s − 7-s − 8-s + 8.06·9-s − 4.15·10-s − 4.14·11-s − 3.32·12-s + 3.57·13-s + 14-s − 13.8·15-s + 16-s − 4.73·17-s − 8.06·18-s − 1.24·19-s + 4.15·20-s + 3.32·21-s + 4.14·22-s + 5.30·23-s + 3.32·24-s + 12.2·25-s − 3.57·26-s − 16.8·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.92·3-s + 0.5·4-s + 1.85·5-s + 1.35·6-s − 0.377·7-s − 0.353·8-s + 2.68·9-s − 1.31·10-s − 1.25·11-s − 0.960·12-s + 0.990·13-s + 0.267·14-s − 3.56·15-s + 0.250·16-s − 1.14·17-s − 1.90·18-s − 0.286·19-s + 0.929·20-s + 0.725·21-s + 0.884·22-s + 1.10·23-s + 0.678·24-s + 2.45·25-s − 0.700·26-s − 3.23·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 5.30T + 23T^{2} \) |
| 29 | \( 1 - 4.77T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 7.59T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 + 8.81T + 67T^{2} \) |
| 71 | \( 1 - 1.65T + 71T^{2} \) |
| 73 | \( 1 - 9.33T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−7.44188861531968928907427912990, −6.62758205806034830002361343414, −6.42286792115392545688758443963, −5.61293862498305830619089623561, −5.28216020440483767150930703713, −4.42334861587004577933229151984, −2.91517479927029143650991776128, −1.91347848334254974973762952194, −1.16502676309428807298464020340, 0,
1.16502676309428807298464020340, 1.91347848334254974973762952194, 2.91517479927029143650991776128, 4.42334861587004577933229151984, 5.28216020440483767150930703713, 5.61293862498305830619089623561, 6.42286792115392545688758443963, 6.62758205806034830002361343414, 7.44188861531968928907427912990
