| L(s) = 1 | + 2-s − 3.36·3-s + 4-s + 2.63·5-s − 3.36·6-s − 0.528·7-s + 8-s + 8.30·9-s + 2.63·10-s − 4.66·11-s − 3.36·12-s − 6.23·13-s − 0.528·14-s − 8.85·15-s + 16-s + 2.97·17-s + 8.30·18-s − 2.91·19-s + 2.63·20-s + 1.77·21-s − 4.66·22-s + 0.143·23-s − 3.36·24-s + 1.92·25-s − 6.23·26-s − 17.8·27-s − 0.528·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.94·3-s + 0.5·4-s + 1.17·5-s − 1.37·6-s − 0.199·7-s + 0.353·8-s + 2.76·9-s + 0.832·10-s − 1.40·11-s − 0.970·12-s − 1.72·13-s − 0.141·14-s − 2.28·15-s + 0.250·16-s + 0.721·17-s + 1.95·18-s − 0.668·19-s + 0.588·20-s + 0.387·21-s − 0.994·22-s + 0.0300·23-s − 0.686·24-s + 0.385·25-s − 1.22·26-s − 3.43·27-s − 0.0999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 3.36T + 3T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 + 0.528T + 7T^{2} \) |
| 11 | \( 1 + 4.66T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 2.97T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 0.143T + 23T^{2} \) |
| 29 | \( 1 + 0.658T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 + 6.00T + 41T^{2} \) |
| 43 | \( 1 + 2.63T + 43T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 + 5.38T + 53T^{2} \) |
| 59 | \( 1 + 7.55T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 - 8.87T + 73T^{2} \) |
| 79 | \( 1 + 3.59T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−9.786206715134725955877600277791, −7.898469634542289023296580883246, −7.09012415800767445199868990101, −6.29178183937500685228752832183, −5.71214393499774421381461807231, −5.00509941062607120685238769829, −4.59958484647103165651706708997, −2.81992101821373790738438849290, −1.70372187746147903846673515763, 0,
1.70372187746147903846673515763, 2.81992101821373790738438849290, 4.59958484647103165651706708997, 5.00509941062607120685238769829, 5.71214393499774421381461807231, 6.29178183937500685228752832183, 7.09012415800767445199868990101, 7.898469634542289023296580883246, 9.786206715134725955877600277791
