| L(s) = 1 | − 2·3-s + 4-s − 3·9-s − 2·12-s − 13-s + 16-s + 6·17-s − 2·23-s + 6·25-s + 14·27-s − 10·29-s − 3·36-s + 2·39-s + 8·43-s − 2·48-s − 5·49-s − 12·51-s − 52-s − 2·53-s + 4·61-s + 64-s + 6·68-s + 4·69-s − 12·75-s − 20·79-s − 4·81-s + 20·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 9-s − 0.577·12-s − 0.277·13-s + 1/4·16-s + 1.45·17-s − 0.417·23-s + 6/5·25-s + 2.69·27-s − 1.85·29-s − 1/2·36-s + 0.320·39-s + 1.21·43-s − 0.288·48-s − 5/7·49-s − 1.68·51-s − 0.138·52-s − 0.274·53-s + 0.512·61-s + 1/8·64-s + 0.727·68-s + 0.481·69-s − 1.38·75-s − 2.25·79-s − 4/9·81-s + 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244036 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244036 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9307361842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9307361842\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.931144923873587015073673694927, −8.408486909011940238981299566497, −8.081403556814365619620991486748, −7.25038064772684706053554512263, −7.23769136560338131219122632388, −6.36375886203681583526027515427, −5.98634322619209752201921634912, −5.59820256126613597041086249059, −5.28069074518443602737935578596, −4.67550199492678466714172566208, −3.86057015395193111701613139713, −3.10975673351384299574625047165, −2.75001081480826495914336737773, −1.72024833938566855701291965497, −0.62500790976805621272166526863,
0.62500790976805621272166526863, 1.72024833938566855701291965497, 2.75001081480826495914336737773, 3.10975673351384299574625047165, 3.86057015395193111701613139713, 4.67550199492678466714172566208, 5.28069074518443602737935578596, 5.59820256126613597041086249059, 5.98634322619209752201921634912, 6.36375886203681583526027515427, 7.23769136560338131219122632388, 7.25038064772684706053554512263, 8.081403556814365619620991486748, 8.408486909011940238981299566497, 8.931144923873587015073673694927
