L(s) = 1 | − 2.39·2-s − 1.65·3-s + 3.72·4-s + 0.754·5-s + 3.95·6-s − 2.75·7-s − 4.13·8-s − 0.263·9-s − 1.80·10-s − 5.38·11-s − 6.16·12-s − 13-s + 6.59·14-s − 1.24·15-s + 2.44·16-s + 3.28·17-s + 0.630·18-s − 1.26·19-s + 2.81·20-s + 4.55·21-s + 12.8·22-s + 3.25·23-s + 6.84·24-s − 4.43·25-s + 2.39·26-s + 5.39·27-s − 10.2·28-s + ⋯ |
L(s) = 1 | − 1.69·2-s − 0.955·3-s + 1.86·4-s + 0.337·5-s + 1.61·6-s − 1.04·7-s − 1.46·8-s − 0.0878·9-s − 0.570·10-s − 1.62·11-s − 1.78·12-s − 0.277·13-s + 1.76·14-s − 0.322·15-s + 0.610·16-s + 0.796·17-s + 0.148·18-s − 0.289·19-s + 0.628·20-s + 0.994·21-s + 2.74·22-s + 0.678·23-s + 1.39·24-s − 0.886·25-s + 0.469·26-s + 1.03·27-s − 1.94·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1971062347\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1971062347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 - 0.754T + 5T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 + 4.44T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 4.69T + 43T^{2} \) |
| 47 | \( 1 + 0.883T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 6.55T + 59T^{2} \) |
| 61 | \( 1 - 3.34T + 61T^{2} \) |
| 67 | \( 1 + 5.19T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 83 | \( 1 - 0.164T + 83T^{2} \) |
| 89 | \( 1 - 4.30T + 89T^{2} \) |
| 97 | \( 1 - 4.01T + 97T^{2} \) |
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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
−10.17270593741219705224802202442, −9.240563780791214873931885163480, −8.385619937346718289794827918309, −7.49760769730236518221236139917, −6.79001136601685552111289883730, −5.83042228612001720736116057011, −5.18500690325760190266353289843, −3.23348454249769444961363191884, −2.12186931221932549261724189462, −0.43285580134938168288722668300,
0.43285580134938168288722668300, 2.12186931221932549261724189462, 3.23348454249769444961363191884, 5.18500690325760190266353289843, 5.83042228612001720736116057011, 6.79001136601685552111289883730, 7.49760769730236518221236139917, 8.385619937346718289794827918309, 9.240563780791214873931885163480, 10.17270593741219705224802202442