L(s) = 1 | − 2.17·2-s + 1.70·3-s + 2.70·4-s − 3.70·6-s − 3.70·7-s − 1.53·8-s − 0.0783·9-s − 0.630·11-s + 4.63·12-s + 4.34·13-s + 8.04·14-s − 2.07·16-s + 1.55·17-s + 0.170·18-s − 5.70·19-s − 6.34·21-s + 1.36·22-s − 6.63·23-s − 2.63·24-s − 9.41·26-s − 5.26·27-s − 10.0·28-s − 29-s − 2.29·31-s + 7.58·32-s − 1.07·33-s − 3.36·34-s + ⋯ |
L(s) = 1 | − 1.53·2-s + 0.986·3-s + 1.35·4-s − 1.51·6-s − 1.40·7-s − 0.544·8-s − 0.0261·9-s − 0.190·11-s + 1.33·12-s + 1.20·13-s + 2.15·14-s − 0.519·16-s + 0.376·17-s + 0.0400·18-s − 1.30·19-s − 1.38·21-s + 0.291·22-s − 1.38·23-s − 0.537·24-s − 1.84·26-s − 1.01·27-s − 1.89·28-s − 0.185·29-s − 0.411·31-s + 1.34·32-s − 0.187·33-s − 0.577·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 + 0.630T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 + 0.921T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−9.695251653506731718427726241777, −9.130133243145384408237140355544, −8.353583980660990997681755975447, −7.85659065789844309137163593884, −6.65984964922790146994784576808, −5.99566264002174420650366845295, −3.98605099827924716245214754995, −3.01456661681702539706483385628, −1.85658148420143389933265899542, 0,
1.85658148420143389933265899542, 3.01456661681702539706483385628, 3.98605099827924716245214754995, 5.99566264002174420650366845295, 6.65984964922790146994784576808, 7.85659065789844309137163593884, 8.353583980660990997681755975447, 9.130133243145384408237140355544, 9.695251653506731718427726241777