| L(s) = 1 | − 0.162·2-s − 3-s − 1.97·4-s − 4.38·5-s + 0.162·6-s + 7-s + 0.644·8-s + 9-s + 0.711·10-s + 1.97·12-s − 1.67·13-s − 0.162·14-s + 4.38·15-s + 3.84·16-s − 6.18·17-s − 0.162·18-s + 3.47·19-s + 8.65·20-s − 21-s + 0.568·23-s − 0.644·24-s + 14.2·25-s + 0.271·26-s − 27-s − 1.97·28-s + 8.86·29-s − 0.711·30-s + ⋯ |
| L(s) = 1 | − 0.114·2-s − 0.577·3-s − 0.986·4-s − 1.96·5-s + 0.0661·6-s + 0.377·7-s + 0.227·8-s + 0.333·9-s + 0.224·10-s + 0.569·12-s − 0.464·13-s − 0.0433·14-s + 1.13·15-s + 0.960·16-s − 1.50·17-s − 0.0382·18-s + 0.797·19-s + 1.93·20-s − 0.218·21-s + 0.118·23-s − 0.131·24-s + 2.84·25-s + 0.0532·26-s − 0.192·27-s − 0.372·28-s + 1.64·29-s − 0.129·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.162T + 2T^{2} \) |
| 5 | \( 1 + 4.38T + 5T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 + 6.18T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 - 0.568T + 23T^{2} \) |
| 29 | \( 1 - 8.86T + 29T^{2} \) |
| 31 | \( 1 - 4.33T + 31T^{2} \) |
| 37 | \( 1 - 0.969T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 - 4.67T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 2.84T + 59T^{2} \) |
| 61 | \( 1 + 9.29T + 61T^{2} \) |
| 67 | \( 1 + 7.14T + 67T^{2} \) |
| 71 | \( 1 + 0.794T + 71T^{2} \) |
| 73 | \( 1 + 2.38T + 73T^{2} \) |
| 79 | \( 1 - 0.670T + 79T^{2} \) |
| 83 | \( 1 - 9.28T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.532124124477532301556728170280, −7.76284403193501248214682001696, −7.22672732288939678062544086975, −6.28060942451239582699475912122, −4.86709382993429814447379727598, −4.68632331550878704774416115714, −3.93076106390047906498620605708, −2.91238006140623700919695563764, −0.985152472622101384869202053108, 0,
0.985152472622101384869202053108, 2.91238006140623700919695563764, 3.93076106390047906498620605708, 4.68632331550878704774416115714, 4.86709382993429814447379727598, 6.28060942451239582699475912122, 7.22672732288939678062544086975, 7.76284403193501248214682001696, 8.532124124477532301556728170280
