| L(s) = 1 | + 2·2-s − 4·4-s − 5·5-s − 24·8-s − 10·10-s − 10·11-s − 80·13-s − 16·16-s − 7·17-s − 113·19-s + 20·20-s − 20·22-s + 81·23-s + 25·25-s − 160·26-s + 220·29-s − 189·31-s + 160·32-s − 14·34-s + 170·37-s − 226·38-s + 120·40-s + 130·41-s + 10·43-s + 40·44-s + 162·46-s − 160·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 0.274·11-s − 1.70·13-s − 1/4·16-s − 0.0998·17-s − 1.36·19-s + 0.223·20-s − 0.193·22-s + 0.734·23-s + 1/5·25-s − 1.20·26-s + 1.40·29-s − 1.09·31-s + 0.883·32-s − 0.0706·34-s + 0.755·37-s − 0.964·38-s + 0.474·40-s + 0.495·41-s + 0.0354·43-s + 0.137·44-s + 0.519·46-s − 0.496·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 10 T + p^{3} T^{2} \) |
| 13 | \( 1 + 80 T + p^{3} T^{2} \) |
| 17 | \( 1 + 7 T + p^{3} T^{2} \) |
| 19 | \( 1 + 113 T + p^{3} T^{2} \) |
| 23 | \( 1 - 81 T + p^{3} T^{2} \) |
| 29 | \( 1 - 220 T + p^{3} T^{2} \) |
| 31 | \( 1 + 189 T + p^{3} T^{2} \) |
| 37 | \( 1 - 170 T + p^{3} T^{2} \) |
| 41 | \( 1 - 130 T + p^{3} T^{2} \) |
| 43 | \( 1 - 10 T + p^{3} T^{2} \) |
| 47 | \( 1 + 160 T + p^{3} T^{2} \) |
| 53 | \( 1 + 631 T + p^{3} T^{2} \) |
| 59 | \( 1 - 560 T + p^{3} T^{2} \) |
| 61 | \( 1 - 229 T + p^{3} T^{2} \) |
| 67 | \( 1 - 750 T + p^{3} T^{2} \) |
| 71 | \( 1 + 890 T + p^{3} T^{2} \) |
| 73 | \( 1 + 890 T + p^{3} T^{2} \) |
| 79 | \( 1 + 27 T + p^{3} T^{2} \) |
| 83 | \( 1 + 429 T + p^{3} T^{2} \) |
| 89 | \( 1 - 750 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1480 T + p^{3} T^{2} \) |
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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
−12.57424566014826645834747926116, −11.48492018631776667250761358847, −10.17597884772070947559802508637, −9.088461282349262110100015889095, −7.956118376674466054785113985274, −6.62927716936187337621733686286, −5.14667699242776163098295302473, −4.30049136594134854324859927555, −2.76553455228323731262272652886, 0,
2.76553455228323731262272652886, 4.30049136594134854324859927555, 5.14667699242776163098295302473, 6.62927716936187337621733686286, 7.956118376674466054785113985274, 9.088461282349262110100015889095, 10.17597884772070947559802508637, 11.48492018631776667250761358847, 12.57424566014826645834747926116
