| L(s) = 1 | − 2.73·2-s − 3-s + 5.45·4-s + 3.67·5-s + 2.73·6-s − 9.43·8-s + 9-s − 10.0·10-s − 5.55·11-s − 5.45·12-s + 1.35·13-s − 3.67·15-s + 14.8·16-s − 5.33·17-s − 2.73·18-s − 3.13·19-s + 20.0·20-s + 15.1·22-s + 6.55·23-s + 9.43·24-s + 8.54·25-s − 3.68·26-s − 27-s − 8.35·29-s + 10.0·30-s + 3.83·31-s − 21.6·32-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.72·4-s + 1.64·5-s + 1.11·6-s − 3.33·8-s + 0.333·9-s − 3.17·10-s − 1.67·11-s − 1.57·12-s + 0.374·13-s − 0.950·15-s + 3.71·16-s − 1.29·17-s − 0.643·18-s − 0.718·19-s + 4.48·20-s + 3.23·22-s + 1.36·23-s + 1.92·24-s + 1.70·25-s − 0.723·26-s − 0.192·27-s − 1.55·29-s + 1.83·30-s + 0.688·31-s − 3.83·32-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
| \(L(1)\) |
\(\approx\) |
\(0.6355004474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6355004474\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 5.33T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 - 6.55T + 23T^{2} \) |
| 29 | \( 1 + 8.35T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 0.883T + 37T^{2} \) |
| 43 | \( 1 + 3.78T + 43T^{2} \) |
| 47 | \( 1 - 2.25T + 47T^{2} \) |
| 53 | \( 1 - 4.13T + 53T^{2} \) |
| 59 | \( 1 + 4.43T + 59T^{2} \) |
| 61 | \( 1 - 4.50T + 61T^{2} \) |
| 67 | \( 1 + 9.88T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 + 0.851T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.296021396582639474932693255704, −7.35803838027780974142346017282, −6.84833191281316267599843344394, −6.08140634040461063210945271989, −5.66276954129952105899974702464, −4.77719854319686839265329154436, −3.01136706257579414277763989660, −2.26767140095634166311480066065, −1.73607674965402101596312387642, −0.56045099364394282235686551923,
0.56045099364394282235686551923, 1.73607674965402101596312387642, 2.26767140095634166311480066065, 3.01136706257579414277763989660, 4.77719854319686839265329154436, 5.66276954129952105899974702464, 6.08140634040461063210945271989, 6.84833191281316267599843344394, 7.35803838027780974142346017282, 8.296021396582639474932693255704
