| L(s) = 1 | − 2.37·2-s + 3-s + 3.62·4-s − 2.37·6-s − 7-s − 3.85·8-s + 9-s − 4.76·11-s + 3.62·12-s − 13-s + 2.37·14-s + 1.88·16-s + 0.661·17-s − 2.37·18-s − 0.146·19-s − 21-s + 11.3·22-s + 6.70·23-s − 3.85·24-s + 2.37·26-s + 27-s − 3.62·28-s + 4.85·29-s − 8.34·31-s + 3.22·32-s − 4.76·33-s − 1.56·34-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.577·3-s + 1.81·4-s − 0.968·6-s − 0.377·7-s − 1.36·8-s + 0.333·9-s − 1.43·11-s + 1.04·12-s − 0.277·13-s + 0.633·14-s + 0.472·16-s + 0.160·17-s − 0.559·18-s − 0.0335·19-s − 0.218·21-s + 2.41·22-s + 1.39·23-s − 0.786·24-s + 0.465·26-s + 0.192·27-s − 0.685·28-s + 0.901·29-s − 1.49·31-s + 0.570·32-s − 0.829·33-s − 0.268·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 17 | \( 1 - 0.661T + 17T^{2} \) |
| 19 | \( 1 + 0.146T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 8.34T + 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 + 5.84T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 0.420T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 - 9.64T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 0.0279T + 73T^{2} \) |
| 79 | \( 1 + 8.81T + 79T^{2} \) |
| 83 | \( 1 - 0.984T + 83T^{2} \) |
| 89 | \( 1 + 3.03T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.79311669396783276555060116962, −7.11972150593996699697566842721, −6.77985801421423099902633214482, −5.57114051543409385001895505400, −4.90661844530449136250253520917, −3.64562669031886076832869304802, −2.74901876437424105001926223920, −2.22398210855133496523612312391, −1.08933024369710836498534256485, 0,
1.08933024369710836498534256485, 2.22398210855133496523612312391, 2.74901876437424105001926223920, 3.64562669031886076832869304802, 4.90661844530449136250253520917, 5.57114051543409385001895505400, 6.77985801421423099902633214482, 7.11972150593996699697566842721, 7.79311669396783276555060116962
