| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 12-s − 6·13-s + 14-s + 16-s − 4·17-s − 18-s + 2·19-s − 21-s − 23-s − 24-s − 5·25-s + 6·26-s + 27-s − 28-s − 2·29-s − 2·31-s − 32-s + 4·34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s − 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.70702594182184, −13.45068925195889, −12.82291875973380, −12.39028910344360, −11.78386837990308, −11.50452132595799, −10.81384421435364, −10.15781184809919, −9.943626162003579, −9.289609631425366, −9.177231201310064, −8.391037060512817, −7.900414455683642, −7.515334869719742, −6.884107392357079, −6.661008952349902, −5.804582052639389, −5.264230385654246, −4.669220222143106, −3.946053569963770, −3.472339919381334, −2.632113587503121, −2.281437598258824, −1.757473614198894, −0.7078212811908110, 0,
0.7078212811908110, 1.757473614198894, 2.281437598258824, 2.632113587503121, 3.472339919381334, 3.946053569963770, 4.669220222143106, 5.264230385654246, 5.804582052639389, 6.661008952349902, 6.884107392357079, 7.515334869719742, 7.900414455683642, 8.391037060512817, 9.177231201310064, 9.289609631425366, 9.943626162003579, 10.15781184809919, 10.81384421435364, 11.50452132595799, 11.78386837990308, 12.39028910344360, 12.82291875973380, 13.45068925195889, 13.70702594182184
