| L(s) = 1 | − 0.167·2-s − 3-s − 1.97·4-s + 0.167·6-s − 4.13·7-s + 0.665·8-s + 9-s − 4.80·11-s + 1.97·12-s − 4·13-s + 0.693·14-s + 3.83·16-s − 5.94·17-s − 0.167·18-s − 19-s + 4.13·21-s + 0.804·22-s + 7.13·23-s − 0.665·24-s + 0.669·26-s − 27-s + 8.16·28-s − 5·29-s + 8.80·31-s − 1.97·32-s + 4.80·33-s + 0.995·34-s + ⋯ |
| L(s) = 1 | − 0.118·2-s − 0.577·3-s − 0.985·4-s + 0.0683·6-s − 1.56·7-s + 0.235·8-s + 0.333·9-s − 1.44·11-s + 0.569·12-s − 1.10·13-s + 0.185·14-s + 0.958·16-s − 1.44·17-s − 0.0394·18-s − 0.229·19-s + 0.903·21-s + 0.171·22-s + 1.48·23-s − 0.135·24-s + 0.131·26-s − 0.192·27-s + 1.54·28-s − 0.928·29-s + 1.58·31-s − 0.348·32-s + 0.836·33-s + 0.170·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2702611496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2702611496\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.167T + 2T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 23 | \( 1 - 7.13T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 - 0.195T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 - 7.46T + 59T^{2} \) |
| 61 | \( 1 + 6.60T + 61T^{2} \) |
| 67 | \( 1 + 1.19T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 + 0.330T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−9.605604055907886670402722554672, −8.938154799313002109138499008504, −7.955343051482554743636463591493, −7.00340272109765059116676305406, −6.32060936102094730119476188868, −5.15532639751351822631500293161, −4.75493741023994062665024257310, −3.48385403646453888532603394780, −2.49511161767803834173530802904, −0.36753422753951969806669635017,
0.36753422753951969806669635017, 2.49511161767803834173530802904, 3.48385403646453888532603394780, 4.75493741023994062665024257310, 5.15532639751351822631500293161, 6.32060936102094730119476188868, 7.00340272109765059116676305406, 7.955343051482554743636463591493, 8.938154799313002109138499008504, 9.605604055907886670402722554672
