| L(s) = 1 | − 0.136·2-s − 0.981·4-s − 1.15·5-s + 0.270·8-s + 9-s + 0.157·10-s + 0.944·16-s − 1.83·17-s − 0.136·18-s + 1.13·20-s + 0.330·25-s − 0.399·32-s + 0.250·34-s − 0.981·36-s + 1.36·37-s − 0.311·40-s + 1.92·43-s − 1.15·45-s + 0.920·47-s + 49-s − 0.0450·50-s + 0.920·61-s − 0.889·64-s − 1.55·67-s + 1.80·68-s + 1.70·71-s + 0.270·72-s + ⋯ |
| L(s) = 1 | − 0.136·2-s − 0.981·4-s − 1.15·5-s + 0.270·8-s + 9-s + 0.157·10-s + 0.944·16-s − 1.83·17-s − 0.136·18-s + 1.13·20-s + 0.330·25-s − 0.399·32-s + 0.250·34-s − 0.981·36-s + 1.36·37-s − 0.311·40-s + 1.92·43-s − 1.15·45-s + 0.920·47-s + 49-s − 0.0450·50-s + 0.920·61-s − 0.889·64-s − 1.55·67-s + 1.80·68-s + 1.70·71-s + 0.270·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6884301830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6884301830\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1+O(T) \) |
| good | 2 | \( 1 + 0.136T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.15T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.83T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.36T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.92T + T^{2} \) |
| 47 | \( 1 - 0.920T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.920T + T^{2} \) |
| 67 | \( 1 + 1.55T + T^{2} \) |
| 71 | \( 1 - 1.70T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.36T + T^{2} \) |
| 89 | \( 1 - 1.92T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.092155244972706812172250868964, −8.252354729325780584475854569995, −7.61947637329322037473716037144, −6.97661651350053265787852736170, −5.95966275243525330389785581194, −4.78655904507191745194851592097, −4.23404007967853980874310024572, −3.78250427411316425530801732407, −2.34968216303399156495913626460, −0.792938904423430949497403974259,
0.792938904423430949497403974259, 2.34968216303399156495913626460, 3.78250427411316425530801732407, 4.23404007967853980874310024572, 4.78655904507191745194851592097, 5.95966275243525330389785581194, 6.97661651350053265787852736170, 7.61947637329322037473716037144, 8.252354729325780584475854569995, 9.092155244972706812172250868964
