| L(s) = 1 | − 2.82·2-s + 5.97·4-s + 0.305·5-s − 1.90·7-s − 11.2·8-s − 0.863·10-s − 0.144·11-s − 1.96·13-s + 5.37·14-s + 19.7·16-s − 3.93·17-s + 6.59·19-s + 1.82·20-s + 0.407·22-s − 5.77·23-s − 4.90·25-s + 5.55·26-s − 11.3·28-s + 5.85·29-s − 2.74·31-s − 33.3·32-s + 11.1·34-s − 0.581·35-s + 10.1·37-s − 18.6·38-s − 3.43·40-s + 4.92·41-s + ⋯ |
| L(s) = 1 | − 1.99·2-s + 2.98·4-s + 0.136·5-s − 0.719·7-s − 3.96·8-s − 0.272·10-s − 0.0435·11-s − 0.545·13-s + 1.43·14-s + 4.93·16-s − 0.954·17-s + 1.51·19-s + 0.408·20-s + 0.0869·22-s − 1.20·23-s − 0.981·25-s + 1.09·26-s − 2.14·28-s + 1.08·29-s − 0.493·31-s − 5.89·32-s + 1.90·34-s − 0.0982·35-s + 1.66·37-s − 3.02·38-s − 0.542·40-s + 0.768·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 2.82T + 2T^{2} \) |
| 5 | \( 1 - 0.305T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 0.144T + 11T^{2} \) |
| 13 | \( 1 + 1.96T + 13T^{2} \) |
| 17 | \( 1 + 3.93T + 17T^{2} \) |
| 19 | \( 1 - 6.59T + 19T^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 8.01T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 - 0.466T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 0.0478T + 71T^{2} \) |
| 73 | \( 1 - 7.71T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 + 1.21T + 89T^{2} \) |
| 97 | \( 1 + 5.11T + 97T^{2} \) |
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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
−7.88035204137970569192149669768, −7.76288468162060123709125754569, −6.83172385530899907042358864123, −6.20082661819711508197541514259, −5.56708602543639966424115143134, −4.00183339652337862262146752387, −2.82771352177254873433244913455, −2.30488204061333930178441828217, −1.09635761184597320413020650093, 0,
1.09635761184597320413020650093, 2.30488204061333930178441828217, 2.82771352177254873433244913455, 4.00183339652337862262146752387, 5.56708602543639966424115143134, 6.20082661819711508197541514259, 6.83172385530899907042358864123, 7.76288468162060123709125754569, 7.88035204137970569192149669768
