| L(s) = 1 | + 4·4-s − 2·7-s + 13-s + 16·16-s + 26·19-s − 8·28-s + 59·31-s + 73·37-s − 83·43-s − 45·49-s + 4·52-s + 74·61-s + 64·64-s + 109·67-s − 143·73-s + 104·76-s + 11·79-s − 2·91-s − 2·97-s + 157·103-s + 71·109-s − 32·112-s + ⋯ |
| L(s) = 1 | + 4-s − 2/7·7-s + 1/13·13-s + 16-s + 1.36·19-s − 2/7·28-s + 1.90·31-s + 1.97·37-s − 1.93·43-s − 0.918·49-s + 1/13·52-s + 1.21·61-s + 64-s + 1.62·67-s − 1.95·73-s + 1.36·76-s + 0.139·79-s − 0.0219·91-s − 0.0206·97-s + 1.52·103-s + 0.651·109-s − 2/7·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.406461869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.406461869\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 2 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 26 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 59 T + p^{2} T^{2} \) |
| 37 | \( 1 - 73 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 83 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 74 T + p^{2} T^{2} \) |
| 67 | \( 1 - 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 143 T + p^{2} T^{2} \) |
| 79 | \( 1 - 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 2 T + p^{2} T^{2} \) |
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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
−10.15161824656917175770300666426, −9.733051588130730517942134686802, −8.399331211301813030974891344490, −7.62484249879596480836156919154, −6.70898555109418166202478111141, −5.98922951754427387857432158616, −4.87416917308249998501226957133, −3.45416788114232643770785061446, −2.55459238372625204128902420036, −1.12004054250425704529292958583,
1.12004054250425704529292958583, 2.55459238372625204128902420036, 3.45416788114232643770785061446, 4.87416917308249998501226957133, 5.98922951754427387857432158616, 6.70898555109418166202478111141, 7.62484249879596480836156919154, 8.399331211301813030974891344490, 9.733051588130730517942134686802, 10.15161824656917175770300666426
