| L(s) = 1 | − 3-s − 3·7-s − 2·9-s + 5·13-s − 7·19-s + 3·21-s − 7·25-s + 5·27-s − 4·31-s + 9·37-s − 5·39-s − 2·43-s − 7·49-s + 7·57-s + 11·61-s + 6·63-s + 10·67-s + 13·73-s + 7·75-s + 9·79-s + 81-s − 15·91-s + 4·93-s − 2·97-s + 25·103-s + 9·109-s − 9·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s + 1.38·13-s − 1.60·19-s + 0.654·21-s − 7/5·25-s + 0.962·27-s − 0.718·31-s + 1.47·37-s − 0.800·39-s − 0.304·43-s − 49-s + 0.927·57-s + 1.40·61-s + 0.755·63-s + 1.22·67-s + 1.52·73-s + 0.808·75-s + 1.01·79-s + 1/9·81-s − 1.57·91-s + 0.414·93-s − 0.203·97-s + 2.46·103-s + 0.862·109-s − 0.854·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.912100337378443845403847680489, −7.40528135448599923369465286739, −6.62486695268191019469988832949, −6.40885283033573869506417577249, −6.22839783717003257680273337335, −5.75673744185712702169778094056, −5.26475584837072496975982767840, −4.69975231950340169326386943461, −4.02258078358775569531789987694, −3.65398622077052143026512221503, −3.26885465455711091297898561237, −2.42295119030324067400978100506, −1.97703736672005995000319523704, −0.864993252576462591471178521392, 0,
0.864993252576462591471178521392, 1.97703736672005995000319523704, 2.42295119030324067400978100506, 3.26885465455711091297898561237, 3.65398622077052143026512221503, 4.02258078358775569531789987694, 4.69975231950340169326386943461, 5.26475584837072496975982767840, 5.75673744185712702169778094056, 6.22839783717003257680273337335, 6.40885283033573869506417577249, 6.62486695268191019469988832949, 7.40528135448599923369465286739, 7.912100337378443845403847680489
