| L(s) = 1 | − 2.20·2-s − 3-s + 2.84·4-s − 2.77·5-s + 2.20·6-s − 1.85·8-s + 9-s + 6.11·10-s − 1.68·11-s − 2.84·12-s − 1.26·13-s + 2.77·15-s − 1.59·16-s + 5.32·17-s − 2.20·18-s + 0.141·19-s − 7.89·20-s + 3.69·22-s − 3.48·23-s + 1.85·24-s + 2.70·25-s + 2.78·26-s − 27-s − 2.99·29-s − 6.11·30-s + 5.66·31-s + 7.23·32-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 0.577·3-s + 1.42·4-s − 1.24·5-s + 0.898·6-s − 0.657·8-s + 0.333·9-s + 1.93·10-s − 0.506·11-s − 0.821·12-s − 0.351·13-s + 0.716·15-s − 0.399·16-s + 1.29·17-s − 0.518·18-s + 0.0325·19-s − 1.76·20-s + 0.788·22-s − 0.725·23-s + 0.379·24-s + 0.541·25-s + 0.547·26-s − 0.192·27-s − 0.555·29-s − 1.11·30-s + 1.01·31-s + 1.27·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2911620676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2911620676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 0.141T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 2.99T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 - 0.881T + 37T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 - 9.47T + 61T^{2} \) |
| 67 | \( 1 - 5.95T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 - 1.94T + 73T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 - 0.172T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 5.03T + 97T^{2} \) |
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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
−8.033469431434217945351013852869, −7.62605405961413789097490012725, −7.03593704607522532549160156446, −6.19168461645921894108291024210, −5.27835276055706392222045456649, −4.43771031398717476975949879560, −3.58066448605882053444033051861, −2.54339040697077006832436748013, −1.36709908960674315782160077571, −0.40015389563776778937661354292,
0.40015389563776778937661354292, 1.36709908960674315782160077571, 2.54339040697077006832436748013, 3.58066448605882053444033051861, 4.43771031398717476975949879560, 5.27835276055706392222045456649, 6.19168461645921894108291024210, 7.03593704607522532549160156446, 7.62605405961413789097490012725, 8.033469431434217945351013852869
