L(s) = 1 | + 1.49·2-s + 0.384·3-s + 0.241·4-s + 2.16·5-s + 0.575·6-s − 2.79·7-s − 2.63·8-s − 2.85·9-s + 3.24·10-s − 11-s + 0.0927·12-s − 4.18·14-s + 0.832·15-s − 4.42·16-s − 3.44·17-s − 4.27·18-s + 0.00264·19-s + 0.523·20-s − 1.07·21-s − 1.49·22-s + 7.04·23-s − 1.01·24-s − 0.299·25-s − 2.24·27-s − 0.674·28-s − 3.93·29-s + 1.24·30-s + ⋯ |
L(s) = 1 | + 1.05·2-s + 0.221·3-s + 0.120·4-s + 0.969·5-s + 0.234·6-s − 1.05·7-s − 0.930·8-s − 0.950·9-s + 1.02·10-s − 0.301·11-s + 0.0267·12-s − 1.11·14-s + 0.215·15-s − 1.10·16-s − 0.834·17-s − 1.00·18-s + 0.000606·19-s + 0.117·20-s − 0.234·21-s − 0.319·22-s + 1.46·23-s − 0.206·24-s − 0.0599·25-s − 0.432·27-s − 0.127·28-s − 0.730·29-s + 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 3 | \( 1 - 0.384T + 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 - 0.00264T + 19T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 - 1.77T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 8.73T + 53T^{2} \) |
| 59 | \( 1 - 5.43T + 59T^{2} \) |
| 61 | \( 1 + 4.14T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 - 0.934T + 71T^{2} \) |
| 73 | \( 1 - 2.93T + 73T^{2} \) |
| 79 | \( 1 - 2.39T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 6.54T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.024809544934492137279484115742, −8.219211090669705210320907394833, −6.73111879234350852278166299249, −6.41190414380979492287899609064, −5.40796079514082816269453749600, −5.01268828787392738602891457427, −3.62557067003437643797314319299, −3.05727681247327257001718026056, −2.12839767088643104061709464304, 0,
2.12839767088643104061709464304, 3.05727681247327257001718026056, 3.62557067003437643797314319299, 5.01268828787392738602891457427, 5.40796079514082816269453749600, 6.41190414380979492287899609064, 6.73111879234350852278166299249, 8.219211090669705210320907394833, 9.024809544934492137279484115742