L(s) = 1 | − 2-s + 3-s + 4-s − 2.64·5-s − 6-s − 1.21·7-s − 8-s + 9-s + 2.64·10-s − 2.84·11-s + 12-s − 4.46·13-s + 1.21·14-s − 2.64·15-s + 16-s + 3.03·17-s − 18-s + 19-s − 2.64·20-s − 1.21·21-s + 2.84·22-s + 4.53·23-s − 24-s + 1.98·25-s + 4.46·26-s + 27-s − 1.21·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.18·5-s − 0.408·6-s − 0.459·7-s − 0.353·8-s + 0.333·9-s + 0.835·10-s − 0.859·11-s + 0.288·12-s − 1.23·13-s + 0.325·14-s − 0.682·15-s + 0.250·16-s + 0.735·17-s − 0.235·18-s + 0.229·19-s − 0.590·20-s − 0.265·21-s + 0.607·22-s + 0.944·23-s − 0.204·24-s + 0.396·25-s + 0.874·26-s + 0.192·27-s − 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 - 3.80T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 - 5.76T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−7.902266923046296269121406476896, −7.22507912899524128864114308603, −6.71790882227632457507181820816, −5.54923170975735364283016535134, −4.74781388801709966351329851328, −3.91961781438839004668608745544, −2.88850296563233167067350665984, −2.64039589117194502619046878370, −1.08420510088756435687541401605, 0,
1.08420510088756435687541401605, 2.64039589117194502619046878370, 2.88850296563233167067350665984, 3.91961781438839004668608745544, 4.74781388801709966351329851328, 5.54923170975735364283016535134, 6.71790882227632457507181820816, 7.22507912899524128864114308603, 7.902266923046296269121406476896