| L(s) = 1 | − 2.48·2-s + 0.221·3-s + 4.17·4-s − 2.50·5-s − 0.550·6-s + 2.33·7-s − 5.39·8-s − 2.95·9-s + 6.23·10-s − 4.94·11-s + 0.924·12-s − 3.44·13-s − 5.78·14-s − 0.555·15-s + 5.05·16-s + 4.24·17-s + 7.32·18-s + 19-s − 10.4·20-s + 0.516·21-s + 12.2·22-s − 3.69·23-s − 1.19·24-s + 1.29·25-s + 8.55·26-s − 1.31·27-s + 9.71·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.127·3-s + 2.08·4-s − 1.12·5-s − 0.224·6-s + 0.880·7-s − 1.90·8-s − 0.983·9-s + 1.97·10-s − 1.49·11-s + 0.266·12-s − 0.955·13-s − 1.54·14-s − 0.143·15-s + 1.26·16-s + 1.02·17-s + 1.72·18-s + 0.229·19-s − 2.33·20-s + 0.112·21-s + 2.61·22-s − 0.771·23-s − 0.243·24-s + 0.258·25-s + 1.67·26-s − 0.253·27-s + 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 3 | \( 1 - 0.221T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 - 8.72T + 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 + 6.41T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 2.69T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 0.0512T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 0.953T + 73T^{2} \) |
| 79 | \( 1 - 0.539T + 79T^{2} \) |
| 83 | \( 1 - 4.69T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−8.145156926677405437259412596419, −7.78991617232036513298706532772, −7.19637448770087144953202116805, −6.02607072344221915185118077237, −5.19241951922625401879598619238, −4.24980434518783803253893330355, −2.80959568329311938252433494731, −2.47715114219567106867810996108, −0.976066840455629509767778549600, 0,
0.976066840455629509767778549600, 2.47715114219567106867810996108, 2.80959568329311938252433494731, 4.24980434518783803253893330355, 5.19241951922625401879598619238, 6.02607072344221915185118077237, 7.19637448770087144953202116805, 7.78991617232036513298706532772, 8.145156926677405437259412596419
