| L(s) = 1 | − 2-s − 3·3-s + 4-s − 5-s + 3·6-s + 7-s − 8-s + 6·9-s + 10-s − 5·11-s − 3·12-s + 4·13-s − 14-s + 3·15-s + 16-s − 2·17-s − 6·18-s + 19-s − 20-s − 3·21-s + 5·22-s + 23-s + 3·24-s − 4·25-s − 4·26-s − 9·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 1.50·11-s − 0.866·12-s + 1.10·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.485·17-s − 1.41·18-s + 0.229·19-s − 0.223·20-s − 0.654·21-s + 1.06·22-s + 0.208·23-s + 0.612·24-s − 4/5·25-s − 0.784·26-s − 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.64884889826413312965307712876, −7.07455490546813048538689248212, −6.19017432482913072542410225140, −5.75399049088145624441719823169, −4.96375396305786129303985212902, −4.32345943867608262068456440306, −3.19769028558420132756287120447, −1.95861767781003005886407142186, −0.910983296152825876203101051518, 0,
0.910983296152825876203101051518, 1.95861767781003005886407142186, 3.19769028558420132756287120447, 4.32345943867608262068456440306, 4.96375396305786129303985212902, 5.75399049088145624441719823169, 6.19017432482913072542410225140, 7.07455490546813048538689248212, 7.64884889826413312965307712876
