L(s) = 1 | − 1.16·2-s − 3-s − 0.641·4-s + 1.16·6-s − 4.15·7-s + 3.07·8-s + 9-s − 3.81·11-s + 0.641·12-s + 4.61·13-s + 4.84·14-s − 2.30·16-s + 1.23·17-s − 1.16·18-s − 2.53·19-s + 4.15·21-s + 4.44·22-s + 2.58·23-s − 3.07·24-s − 5.37·26-s − 27-s + 2.66·28-s − 5.43·29-s − 6.09·31-s − 3.47·32-s + 3.81·33-s − 1.43·34-s + ⋯ |
L(s) = 1 | − 0.824·2-s − 0.577·3-s − 0.320·4-s + 0.475·6-s − 1.57·7-s + 1.08·8-s + 0.333·9-s − 1.14·11-s + 0.185·12-s + 1.27·13-s + 1.29·14-s − 0.576·16-s + 0.298·17-s − 0.274·18-s − 0.580·19-s + 0.907·21-s + 0.947·22-s + 0.538·23-s − 0.628·24-s − 1.05·26-s − 0.192·27-s + 0.504·28-s − 1.00·29-s − 1.09·31-s − 0.613·32-s + 0.663·33-s − 0.245·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 - 1.49T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 - 2.67T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.327247560130964086158499034543, −7.45517375200774207200146770534, −6.87984304621849587461439614898, −5.84993307402755106848826883428, −5.48274140512878366487432870506, −4.21001921348503865625408413327, −3.57649512730387590747779449411, −2.40022823262089500577828197956, −0.963483021507229044700754029748, 0,
0.963483021507229044700754029748, 2.40022823262089500577828197956, 3.57649512730387590747779449411, 4.21001921348503865625408413327, 5.48274140512878366487432870506, 5.84993307402755106848826883428, 6.87984304621849587461439614898, 7.45517375200774207200146770534, 8.327247560130964086158499034543