L(s) = 1 | + 2.31·2-s + 1.34·3-s + 3.37·4-s − 4.07·5-s + 3.12·6-s − 2.77·7-s + 3.18·8-s − 1.18·9-s − 9.44·10-s + 0.752·11-s + 4.54·12-s − 1.72·13-s − 6.44·14-s − 5.49·15-s + 0.633·16-s − 3.25·17-s − 2.73·18-s − 0.0478·19-s − 13.7·20-s − 3.74·21-s + 1.74·22-s + 2.03·23-s + 4.29·24-s + 11.5·25-s − 3.98·26-s − 5.63·27-s − 9.37·28-s + ⋯ |
L(s) = 1 | + 1.63·2-s + 0.778·3-s + 1.68·4-s − 1.82·5-s + 1.27·6-s − 1.05·7-s + 1.12·8-s − 0.393·9-s − 2.98·10-s + 0.227·11-s + 1.31·12-s − 0.477·13-s − 1.72·14-s − 1.41·15-s + 0.158·16-s − 0.788·17-s − 0.645·18-s − 0.0109·19-s − 3.07·20-s − 0.817·21-s + 0.372·22-s + 0.425·23-s + 0.876·24-s + 2.31·25-s − 0.782·26-s − 1.08·27-s − 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 - 0.752T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 + 3.25T + 17T^{2} \) |
| 19 | \( 1 + 0.0478T + 19T^{2} \) |
| 23 | \( 1 - 2.03T + 23T^{2} \) |
| 29 | \( 1 - 0.747T + 29T^{2} \) |
| 31 | \( 1 + 3.44T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 - 8.40T + 41T^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 - 1.78T + 53T^{2} \) |
| 59 | \( 1 - 5.78T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 + 7.27T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 5.67T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.738372463804892970834685398118, −7.893860112288655520875812305124, −7.05485795625110249785234005041, −6.54069727143699594530721519484, −5.36958622357276935482621560486, −4.46056296224328290020878701600, −3.69710775641180998224277578013, −3.26773885408604979425528548169, −2.44348163714093411898305892893, 0,
2.44348163714093411898305892893, 3.26773885408604979425528548169, 3.69710775641180998224277578013, 4.46056296224328290020878701600, 5.36958622357276935482621560486, 6.54069727143699594530721519484, 7.05485795625110249785234005041, 7.893860112288655520875812305124, 8.738372463804892970834685398118