| L(s) = 1 | + 2.03·2-s − 0.747·3-s + 2.14·4-s − 1.25·5-s − 1.52·6-s + 0.391·7-s + 0.288·8-s − 2.44·9-s − 2.56·10-s − 4.83·11-s − 1.60·12-s − 1.03·13-s + 0.797·14-s + 0.940·15-s − 3.69·16-s − 17-s − 4.96·18-s + 0.340·19-s − 2.69·20-s − 0.292·21-s − 9.83·22-s − 5.15·23-s − 0.215·24-s − 3.41·25-s − 2.09·26-s + 4.06·27-s + 0.838·28-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.431·3-s + 1.07·4-s − 0.563·5-s − 0.620·6-s + 0.148·7-s + 0.102·8-s − 0.813·9-s − 0.810·10-s − 1.45·11-s − 0.461·12-s − 0.285·13-s + 0.213·14-s + 0.242·15-s − 0.924·16-s − 0.242·17-s − 1.17·18-s + 0.0780·19-s − 0.602·20-s − 0.0638·21-s − 2.09·22-s − 1.07·23-s − 0.0440·24-s − 0.682·25-s − 0.411·26-s + 0.782·27-s + 0.158·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 3 | \( 1 + 0.747T + 3T^{2} \) |
| 5 | \( 1 + 1.25T + 5T^{2} \) |
| 7 | \( 1 - 0.391T + 7T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 + 1.03T + 13T^{2} \) |
| 19 | \( 1 - 0.340T + 19T^{2} \) |
| 23 | \( 1 + 5.15T + 23T^{2} \) |
| 29 | \( 1 - 7.62T + 29T^{2} \) |
| 31 | \( 1 - 9.77T + 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 + 0.809T + 53T^{2} \) |
| 61 | \( 1 + 9.58T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 2.93T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 - 6.90T + 79T^{2} \) |
| 83 | \( 1 - 0.141T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−9.819281468939024163604598525703, −8.363995384727490989649050757397, −7.915383166146715244298672441074, −6.59038554180961687271154738840, −5.93743219587656079088967312770, −5.00261022775202690097557473626, −4.50432222041827208048138175602, −3.23275800744996616290562374392, −2.47466471512006846775968953370, 0,
2.47466471512006846775968953370, 3.23275800744996616290562374392, 4.50432222041827208048138175602, 5.00261022775202690097557473626, 5.93743219587656079088967312770, 6.59038554180961687271154738840, 7.915383166146715244298672441074, 8.363995384727490989649050757397, 9.819281468939024163604598525703
