| L(s) = 1 | − 3·3-s + 6·9-s + 3·13-s + 19-s + 3·25-s − 9·27-s − 12·31-s − 6·37-s − 9·39-s + 5·43-s − 5·49-s − 3·57-s + 17·61-s − 3·67-s + 9·73-s − 9·75-s − 9·79-s + 9·81-s + 36·93-s + 6·97-s − 6·109-s + 18·111-s + 18·117-s + 6·121-s + 127-s − 15·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 0.832·13-s + 0.229·19-s + 3/5·25-s − 1.73·27-s − 2.15·31-s − 0.986·37-s − 1.44·39-s + 0.762·43-s − 5/7·49-s − 0.397·57-s + 2.17·61-s − 0.366·67-s + 1.05·73-s − 1.03·75-s − 1.01·79-s + 81-s + 3.73·93-s + 0.609·97-s − 0.574·109-s + 1.70·111-s + 1.66·117-s + 6/11·121-s + 0.0887·127-s − 1.32·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 21 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 63 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 57 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.65478542993864032991661837675, −7.36028396690067987832949372847, −6.97962741649462758989451840867, −6.42454560680663576093867154103, −6.20130789958900120032074211076, −5.60043213822687702964922228729, −5.22610279522421829320276692403, −5.02175821113324799428203151522, −4.24507010736908696665397610267, −3.80392351122560825863494767467, −3.36903337610250286320755243958, −2.40344357012843977113173240461, −1.62531368494710710101103096307, −0.999909831714497685598503526662, 0,
0.999909831714497685598503526662, 1.62531368494710710101103096307, 2.40344357012843977113173240461, 3.36903337610250286320755243958, 3.80392351122560825863494767467, 4.24507010736908696665397610267, 5.02175821113324799428203151522, 5.22610279522421829320276692403, 5.60043213822687702964922228729, 6.20130789958900120032074211076, 6.42454560680663576093867154103, 6.97962741649462758989451840867, 7.36028396690067987832949372847, 7.65478542993864032991661837675
