| L(s) = 1 | − 0.392·2-s + 0.950·3-s − 1.84·4-s − 2.28·5-s − 0.372·6-s + 2.71·7-s + 1.50·8-s − 2.09·9-s + 0.897·10-s + 1.36·11-s − 1.75·12-s − 2.93·13-s − 1.06·14-s − 2.17·15-s + 3.10·16-s − 2.61·17-s + 0.822·18-s − 7.79·19-s + 4.22·20-s + 2.57·21-s − 0.536·22-s − 2.61·23-s + 1.43·24-s + 0.230·25-s + 1.14·26-s − 4.84·27-s − 5.00·28-s + ⋯ |
| L(s) = 1 | − 0.277·2-s + 0.548·3-s − 0.923·4-s − 1.02·5-s − 0.152·6-s + 1.02·7-s + 0.533·8-s − 0.699·9-s + 0.283·10-s + 0.412·11-s − 0.506·12-s − 0.812·13-s − 0.284·14-s − 0.561·15-s + 0.775·16-s − 0.633·17-s + 0.193·18-s − 1.78·19-s + 0.944·20-s + 0.561·21-s − 0.114·22-s − 0.544·23-s + 0.292·24-s + 0.0460·25-s + 0.225·26-s − 0.932·27-s − 0.945·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 503 | \( 1 + T \) |
| good | 2 | \( 1 + 0.392T + 2T^{2} \) |
| 3 | \( 1 - 0.950T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 + 7.79T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 + 0.314T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 + 4.17T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 - 0.457T + 43T^{2} \) |
| 47 | \( 1 - 7.67T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 - 0.217T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 0.0952T + 89T^{2} \) |
| 97 | \( 1 + 9.27T + 97T^{2} \) |
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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
−10.53010239666383339771505347511, −9.260756767745355465147802581238, −8.591021552571008166664217770737, −8.066189830981265060248399066376, −7.21364188783771902971732629678, −5.60490906735082322958828048930, −4.43493388878886222689102822589, −3.85280977327434200316008754894, −2.14541617555194133492562465260, 0,
2.14541617555194133492562465260, 3.85280977327434200316008754894, 4.43493388878886222689102822589, 5.60490906735082322958828048930, 7.21364188783771902971732629678, 8.066189830981265060248399066376, 8.591021552571008166664217770737, 9.260756767745355465147802581238, 10.53010239666383339771505347511
