| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 3·7-s + 8-s + 9-s + 2·10-s + 6·11-s − 12-s − 13-s − 3·14-s − 2·15-s + 16-s − 17-s + 18-s − 19-s + 2·20-s + 3·21-s + 6·22-s − 8·23-s − 24-s − 25-s − 26-s − 27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.80·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s + 0.654·21-s + 1.27·22-s − 1.66·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.68244793344584, −14.45110114911028, −13.85570245383452, −13.44000578271879, −12.90008075300447, −12.35115519319366, −11.92447599398025, −11.58833351261954, −10.75625776303675, −10.28570674646697, −9.600166867895484, −9.502727665338030, −8.758641350822910, −7.949076266600551, −7.110306665043141, −6.635478044766432, −6.283352369221728, −5.821164713885788, −5.342982905644015, −4.298213996024256, −4.046973171964356, −3.388353539182421, −2.476769579037923, −1.882534722849429, −1.119679626868832, 0,
1.119679626868832, 1.882534722849429, 2.476769579037923, 3.388353539182421, 4.046973171964356, 4.298213996024256, 5.342982905644015, 5.821164713885788, 6.283352369221728, 6.635478044766432, 7.110306665043141, 7.949076266600551, 8.758641350822910, 9.502727665338030, 9.600166867895484, 10.28570674646697, 10.75625776303675, 11.58833351261954, 11.92447599398025, 12.35115519319366, 12.90008075300447, 13.44000578271879, 13.85570245383452, 14.45110114911028, 14.68244793344584
