| L(s) = 1 | − 2.29·2-s − 3-s + 3.26·4-s + 2.29·6-s − 4.13·7-s − 2.90·8-s + 9-s − 1.15·11-s − 3.26·12-s + 1.65·13-s + 9.49·14-s + 0.135·16-s + 1.91·17-s − 2.29·18-s + 1.91·19-s + 4.13·21-s + 2.64·22-s − 7.97·23-s + 2.90·24-s − 3.79·26-s − 27-s − 13.5·28-s + 5.49·29-s − 2.62·31-s + 5.50·32-s + 1.15·33-s − 4.38·34-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.63·4-s + 0.936·6-s − 1.56·7-s − 1.02·8-s + 0.333·9-s − 0.347·11-s − 0.942·12-s + 0.459·13-s + 2.53·14-s + 0.0337·16-s + 0.463·17-s − 0.540·18-s + 0.438·19-s + 0.902·21-s + 0.563·22-s − 1.66·23-s + 0.593·24-s − 0.744·26-s − 0.192·27-s − 2.55·28-s + 1.01·29-s − 0.471·31-s + 0.972·32-s + 0.200·33-s − 0.751·34-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 | \( 1 + 2.29T + 2T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 - 1.91T + 19T^{2} \) |
| 23 | \( 1 + 7.97T + 23T^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 0.593T + 37T^{2} \) |
| 41 | \( 1 - 4.60T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 - 1.04T + 67T^{2} \) |
| 71 | \( 1 + 7.46T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 0.0744T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 + 0.130T + 97T^{2} \) |
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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
−8.322542616288997272593805842406, −7.51446042246870503260017771951, −6.88046740454428082532361278176, −6.18563739454805039685567111816, −5.59509125150426359207841715733, −4.22056126736639867593381180620, −3.23560097062500858073011746385, −2.22935116117494944252322498567, −0.966104317192236373178499369673, 0,
0.966104317192236373178499369673, 2.22935116117494944252322498567, 3.23560097062500858073011746385, 4.22056126736639867593381180620, 5.59509125150426359207841715733, 6.18563739454805039685567111816, 6.88046740454428082532361278176, 7.51446042246870503260017771951, 8.322542616288997272593805842406
