L(s) = 1 | − 2-s + 4-s + 0.854·5-s + 0.727·7-s − 8-s − 0.854·10-s − 4.77·11-s + 0.670·13-s − 0.727·14-s + 16-s + 0.231·17-s − 5.18·19-s + 0.854·20-s + 4.77·22-s + 7.77·23-s − 4.27·25-s − 0.670·26-s + 0.727·28-s + 7.62·29-s − 3.13·31-s − 32-s − 0.231·34-s + 0.621·35-s − 2.35·37-s + 5.18·38-s − 0.854·40-s − 0.350·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.382·5-s + 0.274·7-s − 0.353·8-s − 0.270·10-s − 1.44·11-s + 0.185·13-s − 0.194·14-s + 0.250·16-s + 0.0561·17-s − 1.18·19-s + 0.191·20-s + 1.01·22-s + 1.62·23-s − 0.854·25-s − 0.131·26-s + 0.137·28-s + 1.41·29-s − 0.563·31-s − 0.176·32-s − 0.0397·34-s + 0.104·35-s − 0.387·37-s + 0.840·38-s − 0.135·40-s − 0.0547·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 0.854T + 5T^{2} \) |
| 7 | \( 1 - 0.727T + 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 - 0.670T + 13T^{2} \) |
| 17 | \( 1 - 0.231T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 - 7.77T + 23T^{2} \) |
| 29 | \( 1 - 7.62T + 29T^{2} \) |
| 31 | \( 1 + 3.13T + 31T^{2} \) |
| 37 | \( 1 + 2.35T + 37T^{2} \) |
| 41 | \( 1 + 0.350T + 41T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 7.38T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 9.83T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 + 7.56T + 79T^{2} \) |
| 83 | \( 1 + 0.603T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 + 4.56T + 97T^{2} \) |
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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
−7.59046931073615174064731647277, −6.87970623711748285237872978391, −6.23499261086751773824379461863, −5.36898808193926282460315616396, −4.88594331010947757407140075915, −3.81651101884186651690171287990, −2.77259121067442122143927229856, −2.24530393274824927819934120087, −1.19596788733239721808384118667, 0,
1.19596788733239721808384118667, 2.24530393274824927819934120087, 2.77259121067442122143927229856, 3.81651101884186651690171287990, 4.88594331010947757407140075915, 5.36898808193926282460315616396, 6.23499261086751773824379461863, 6.87970623711748285237872978391, 7.59046931073615174064731647277