| L(s) = 1 | − 0.567·2-s − 2.88·3-s − 1.67·4-s + 1.63·6-s − 5.00·7-s + 2.08·8-s + 5.32·9-s + 4.17·11-s + 4.84·12-s + 0.657·13-s + 2.84·14-s + 2.16·16-s − 3.02·18-s − 5.62·19-s + 14.4·21-s − 2.37·22-s − 4.27·23-s − 6.02·24-s − 0.373·26-s − 6.71·27-s + 8.40·28-s − 5.59·29-s + 0.143·31-s − 5.40·32-s − 12.0·33-s − 8.93·36-s − 4.88·37-s + ⋯ |
| L(s) = 1 | − 0.401·2-s − 1.66·3-s − 0.838·4-s + 0.669·6-s − 1.89·7-s + 0.738·8-s + 1.77·9-s + 1.25·11-s + 1.39·12-s + 0.182·13-s + 0.760·14-s + 0.542·16-s − 0.713·18-s − 1.28·19-s + 3.15·21-s − 0.505·22-s − 0.892·23-s − 1.23·24-s − 0.0732·26-s − 1.29·27-s + 1.58·28-s − 1.03·29-s + 0.0258·31-s − 0.956·32-s − 2.09·33-s − 1.48·36-s − 0.803·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.567T + 2T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 7 | \( 1 + 5.00T + 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 - 0.657T + 13T^{2} \) |
| 19 | \( 1 + 5.62T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 + 5.59T + 29T^{2} \) |
| 31 | \( 1 - 0.143T + 31T^{2} \) |
| 37 | \( 1 + 4.88T + 37T^{2} \) |
| 41 | \( 1 + 1.55T + 41T^{2} \) |
| 43 | \( 1 + 2.97T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 0.961T + 61T^{2} \) |
| 67 | \( 1 - 0.747T + 67T^{2} \) |
| 71 | \( 1 - 0.0687T + 71T^{2} \) |
| 73 | \( 1 + 1.46T + 73T^{2} \) |
| 79 | \( 1 - 6.72T + 79T^{2} \) |
| 83 | \( 1 - 6.20T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−7.28720515462880064215200515436, −6.64983015903704041987460493020, −6.24507448900286901777486608976, −5.64962478991780222396187505184, −4.80025967630087028412037389536, −3.88936611252361027400858986371, −3.62296912796068090782005192178, −1.90584200732477861356459808590, −0.71896903901915469068374365289, 0,
0.71896903901915469068374365289, 1.90584200732477861356459808590, 3.62296912796068090782005192178, 3.88936611252361027400858986371, 4.80025967630087028412037389536, 5.64962478991780222396187505184, 6.24507448900286901777486608976, 6.64983015903704041987460493020, 7.28720515462880064215200515436
