| L(s) = 1 | − 3-s + 2·4-s − 3·9-s − 2·12-s − 4·13-s + 2·17-s + 9·23-s − 4·25-s + 4·27-s − 3·29-s − 6·36-s + 4·39-s − 14·43-s − 4·49-s − 2·51-s − 8·52-s + 12·53-s + 13·61-s − 8·64-s + 4·68-s − 9·69-s + 4·75-s + 16·79-s + 2·81-s + 3·87-s + 18·92-s − 8·100-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s − 9-s − 0.577·12-s − 1.10·13-s + 0.485·17-s + 1.87·23-s − 4/5·25-s + 0.769·27-s − 0.557·29-s − 36-s + 0.640·39-s − 2.13·43-s − 4/7·49-s − 0.280·51-s − 1.10·52-s + 1.64·53-s + 1.66·61-s − 64-s + 0.485·68-s − 1.08·69-s + 0.461·75-s + 1.80·79-s + 2/9·81-s + 0.321·87-s + 1.87·92-s − 4/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2873 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2873 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6855048838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6855048838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 175 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
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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
−12.79304833028138192328504218282, −12.02981672125278947513561978236, −11.46359144154241782604076330679, −11.44285898812997260496679827262, −10.56393351695548970570884614107, −9.984183409123755645886076266483, −9.196013450504782341394314060825, −8.476654870964960748666794566098, −7.60959774921550994826949003715, −6.98870454785733725021415544295, −6.38324472864305313284832940816, −5.46558217525874298674502660398, −4.95568921607729631168993937193, −3.38826098191495753735585531973, −2.35417954344122087489804327096,
2.35417954344122087489804327096, 3.38826098191495753735585531973, 4.95568921607729631168993937193, 5.46558217525874298674502660398, 6.38324472864305313284832940816, 6.98870454785733725021415544295, 7.60959774921550994826949003715, 8.476654870964960748666794566098, 9.196013450504782341394314060825, 9.984183409123755645886076266483, 10.56393351695548970570884614107, 11.44285898812997260496679827262, 11.46359144154241782604076330679, 12.02981672125278947513561978236, 12.79304833028138192328504218282
