L(s) = 1 | − 2.61·2-s − 2.82·3-s + 4.86·4-s − 3.49·5-s + 7.38·6-s + 2.54·7-s − 7.49·8-s + 4.95·9-s + 9.15·10-s + 3.23·11-s − 13.7·12-s + 5.10·13-s − 6.65·14-s + 9.85·15-s + 9.91·16-s + 3.26·17-s − 12.9·18-s + 19-s − 16.9·20-s − 7.16·21-s − 8.47·22-s + 0.140·23-s + 21.1·24-s + 7.21·25-s − 13.3·26-s − 5.51·27-s + 12.3·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s − 1.62·3-s + 2.43·4-s − 1.56·5-s + 3.01·6-s + 0.960·7-s − 2.65·8-s + 1.65·9-s + 2.89·10-s + 0.974·11-s − 3.95·12-s + 1.41·13-s − 1.77·14-s + 2.54·15-s + 2.47·16-s + 0.791·17-s − 3.05·18-s + 0.229·19-s − 3.79·20-s − 1.56·21-s − 1.80·22-s + 0.0292·23-s + 4.31·24-s + 1.44·25-s − 2.62·26-s − 1.06·27-s + 2.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 23 | \( 1 - 0.140T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 1.59T + 37T^{2} \) |
| 41 | \( 1 - 0.548T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 0.497T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 4.71T + 73T^{2} \) |
| 79 | \( 1 - 0.388T + 79T^{2} \) |
| 83 | \( 1 + 2.18T + 83T^{2} \) |
| 89 | \( 1 + 4.22T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−8.129595847976677843142655315713, −7.34236220599702311708759628415, −7.02216574828984371194643877137, −6.05938433839272993038254873553, −5.40929944023138614583809335282, −4.19261559506862988146304360804, −3.49291286026733010592952415508, −1.59617962991518178376442466402, −1.03508450293117936772082879817, 0,
1.03508450293117936772082879817, 1.59617962991518178376442466402, 3.49291286026733010592952415508, 4.19261559506862988146304360804, 5.40929944023138614583809335282, 6.05938433839272993038254873553, 7.02216574828984371194643877137, 7.34236220599702311708759628415, 8.129595847976677843142655315713