L(s) = 1 | − 1.77·2-s − 3-s + 1.14·4-s + 1.77·6-s + 1.50·8-s + 9-s − 1.21·11-s − 1.14·12-s + 1.62·13-s − 4.97·16-s + 5.33·17-s − 1.77·18-s − 6.22·19-s + 2.14·22-s + 1.31·23-s − 1.50·24-s − 2.88·26-s − 27-s − 3.80·29-s + 2.85·31-s + 5.81·32-s + 1.21·33-s − 9.47·34-s + 1.14·36-s − 3.37·37-s + 11.0·38-s − 1.62·39-s + ⋯ |
L(s) = 1 | − 1.25·2-s − 0.577·3-s + 0.574·4-s + 0.724·6-s + 0.533·8-s + 0.333·9-s − 0.365·11-s − 0.331·12-s + 0.450·13-s − 1.24·16-s + 1.29·17-s − 0.418·18-s − 1.42·19-s + 0.458·22-s + 0.274·23-s − 0.308·24-s − 0.565·26-s − 0.192·27-s − 0.707·29-s + 0.512·31-s + 1.02·32-s + 0.210·33-s − 1.62·34-s + 0.191·36-s − 0.555·37-s + 1.79·38-s − 0.260·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 + 3.37T + 37T^{2} \) |
| 41 | \( 1 + 2.98T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.87T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 + 3.03T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
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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
−8.019127631616504963954618224753, −7.80045423453204087050357275295, −6.76469059786224860155778458837, −6.14000833818448904470713911038, −5.18875248886568765289297519091, −4.42402222018586755656291768336, −3.39112111129053380714531650259, −2.08072426008667777999553912007, −1.13679310014917961544024666514, 0,
1.13679310014917961544024666514, 2.08072426008667777999553912007, 3.39112111129053380714531650259, 4.42402222018586755656291768336, 5.18875248886568765289297519091, 6.14000833818448904470713911038, 6.76469059786224860155778458837, 7.80045423453204087050357275295, 8.019127631616504963954618224753