| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s − 6·13-s + 16-s − 14·17-s + 2·23-s + 9·25-s − 4·27-s − 2·29-s − 3·36-s + 12·39-s − 10·43-s − 2·48-s + 28·51-s + 6·52-s − 12·53-s + 26·61-s − 64-s + 14·68-s − 4·69-s − 18·75-s − 24·79-s + 5·81-s + 4·87-s − 2·92-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s − 1.66·13-s + 1/4·16-s − 3.39·17-s + 0.417·23-s + 9/5·25-s − 0.769·27-s − 0.371·29-s − 1/2·36-s + 1.92·39-s − 1.52·43-s − 0.288·48-s + 3.92·51-s + 0.832·52-s − 1.64·53-s + 3.32·61-s − 1/8·64-s + 1.69·68-s − 0.481·69-s − 2.07·75-s − 2.70·79-s + 5/9·81-s + 0.428·87-s − 0.208·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06535083924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06535083924\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.651778179157289819042754179318, −8.376975930789174738250102845067, −8.082222482132109528587908771511, −7.20414748838907514408830424100, −7.11234220890101280371900267171, −6.91284153487334545829682759258, −6.48874212048975702864790789889, −6.24527560469835591625648185411, −5.50715091907045947918358528204, −5.24984449593221129434591017306, −4.79154301357126603320122550906, −4.75602597361863509500376179575, −4.09784159482005571172819105668, −4.08478408688125329968607333917, −3.03859581644279290224801464459, −2.79392930467795604649916398065, −2.12668634279296366397398830420, −1.78707377797406110995001512741, −0.919046021218004238441872688475, −0.097638151739995180537399319957,
0.097638151739995180537399319957, 0.919046021218004238441872688475, 1.78707377797406110995001512741, 2.12668634279296366397398830420, 2.79392930467795604649916398065, 3.03859581644279290224801464459, 4.08478408688125329968607333917, 4.09784159482005571172819105668, 4.75602597361863509500376179575, 4.79154301357126603320122550906, 5.24984449593221129434591017306, 5.50715091907045947918358528204, 6.24527560469835591625648185411, 6.48874212048975702864790789889, 6.91284153487334545829682759258, 7.11234220890101280371900267171, 7.20414748838907514408830424100, 8.082222482132109528587908771511, 8.376975930789174738250102845067, 8.651778179157289819042754179318
