L(s) = 1 | − 2-s + 1.69·3-s + 4-s − 4.04·5-s − 1.69·6-s + 7-s − 8-s − 0.129·9-s + 4.04·10-s − 1.22·11-s + 1.69·12-s − 1.01·13-s − 14-s − 6.86·15-s + 16-s + 0.455·17-s + 0.129·18-s + 3.36·19-s − 4.04·20-s + 1.69·21-s + 1.22·22-s − 0.370·23-s − 1.69·24-s + 11.4·25-s + 1.01·26-s − 5.30·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.978·3-s + 0.5·4-s − 1.81·5-s − 0.691·6-s + 0.377·7-s − 0.353·8-s − 0.0430·9-s + 1.28·10-s − 0.370·11-s + 0.489·12-s − 0.280·13-s − 0.267·14-s − 1.77·15-s + 0.250·16-s + 0.110·17-s + 0.0304·18-s + 0.772·19-s − 0.905·20-s + 0.369·21-s + 0.261·22-s − 0.0771·23-s − 0.345·24-s + 2.28·25-s + 0.198·26-s − 1.02·27-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 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 - 0.455T + 17T^{2} \) |
| 19 | \( 1 - 3.36T + 19T^{2} \) |
| 23 | \( 1 + 0.370T + 23T^{2} \) |
| 29 | \( 1 - 3.63T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 - 3.64T + 41T^{2} \) |
| 43 | \( 1 - 7.02T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 5.61T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.79796734821284933463969468234, −7.46570490014225560225941691198, −6.71298092067878237285604450162, −5.53780876778824635543243763131, −4.62275493417973632195580882550, −3.86088535398185499039878000963, −3.08570578207184230554881998331, −2.53032985974689985710742745299, −1.15685597988632827491185201558, 0,
1.15685597988632827491185201558, 2.53032985974689985710742745299, 3.08570578207184230554881998331, 3.86088535398185499039878000963, 4.62275493417973632195580882550, 5.53780876778824635543243763131, 6.71298092067878237285604450162, 7.46570490014225560225941691198, 7.79796734821284933463969468234