| L(s) = 1 | + 2.36·2-s − 3-s + 3.60·4-s − 1.54·5-s − 2.36·6-s + 7-s + 3.79·8-s + 9-s − 3.64·10-s − 0.663·11-s − 3.60·12-s − 1.26·13-s + 2.36·14-s + 1.54·15-s + 1.77·16-s − 3.21·17-s + 2.36·18-s + 6.63·19-s − 5.55·20-s − 21-s − 1.57·22-s − 4.90·23-s − 3.79·24-s − 2.62·25-s − 2.98·26-s − 27-s + 3.60·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.577·3-s + 1.80·4-s − 0.689·5-s − 0.966·6-s + 0.377·7-s + 1.34·8-s + 0.333·9-s − 1.15·10-s − 0.200·11-s − 1.04·12-s − 0.349·13-s + 0.632·14-s + 0.397·15-s + 0.444·16-s − 0.779·17-s + 0.557·18-s + 1.52·19-s − 1.24·20-s − 0.218·21-s − 0.334·22-s − 1.02·23-s − 0.774·24-s − 0.525·25-s − 0.585·26-s − 0.192·27-s + 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 11 | \( 1 + 0.663T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 + 2.00T + 37T^{2} \) |
| 41 | \( 1 - 2.06T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 + 6.81T + 53T^{2} \) |
| 59 | \( 1 - 1.90T + 59T^{2} \) |
| 61 | \( 1 + 4.61T + 61T^{2} \) |
| 67 | \( 1 - 9.22T + 67T^{2} \) |
| 71 | \( 1 + 8.97T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 0.0721T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 4.20T + 97T^{2} \) |
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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
−7.31056503704410864486122416357, −6.55386007356573654679882343539, −5.93356080428534507105548149247, −5.19997234276446451436688146404, −4.71388256476192073093556452945, −4.07581058358430882715129097573, −3.37157534592670101396063631069, −2.53962007463274417889758488299, −1.53267187749875286444036709497, 0,
1.53267187749875286444036709497, 2.53962007463274417889758488299, 3.37157534592670101396063631069, 4.07581058358430882715129097573, 4.71388256476192073093556452945, 5.19997234276446451436688146404, 5.93356080428534507105548149247, 6.55386007356573654679882343539, 7.31056503704410864486122416357
