L(s) = 1 | − 2-s + 2.60·3-s + 4-s + 1.05·5-s − 2.60·6-s + 3.90·7-s − 8-s + 3.76·9-s − 1.05·10-s − 1.93·11-s + 2.60·12-s + 2.83·13-s − 3.90·14-s + 2.73·15-s + 16-s + 3.73·17-s − 3.76·18-s − 1.04·19-s + 1.05·20-s + 10.1·21-s + 1.93·22-s − 0.280·23-s − 2.60·24-s − 3.89·25-s − 2.83·26-s + 1.98·27-s + 3.90·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.50·3-s + 0.5·4-s + 0.469·5-s − 1.06·6-s + 1.47·7-s − 0.353·8-s + 1.25·9-s − 0.332·10-s − 0.584·11-s + 0.750·12-s + 0.787·13-s − 1.04·14-s + 0.705·15-s + 0.250·16-s + 0.906·17-s − 0.886·18-s − 0.240·19-s + 0.234·20-s + 2.21·21-s + 0.413·22-s − 0.0583·23-s − 0.530·24-s − 0.779·25-s − 0.556·26-s + 0.381·27-s + 0.738·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.336806608\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.336806608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 1.05T + 5T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 + 1.93T + 11T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 23 | \( 1 + 0.280T + 23T^{2} \) |
| 29 | \( 1 - 9.47T + 29T^{2} \) |
| 31 | \( 1 + 0.699T + 31T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 - 6.13T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 + 4.58T + 67T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 - 0.806T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 5.03T + 83T^{2} \) |
| 89 | \( 1 - 6.52T + 89T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
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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
−8.572989999139362327207618885683, −7.85460947863122318690079375466, −7.53054167697114578774478513791, −6.38844121397986411219600768203, −5.49200655496347961051319995371, −4.59192970789156247740743340584, −3.59631332880332225953570559128, −2.71078924656190184127211258141, −1.94069632044529288563475710367, −1.21592926220285357129408492626,
1.21592926220285357129408492626, 1.94069632044529288563475710367, 2.71078924656190184127211258141, 3.59631332880332225953570559128, 4.59192970789156247740743340584, 5.49200655496347961051319995371, 6.38844121397986411219600768203, 7.53054167697114578774478513791, 7.85460947863122318690079375466, 8.572989999139362327207618885683