| L(s) = 1 | − 3-s − 2·4-s − 2·9-s + 2·12-s + 4·13-s − 4·17-s − 3·23-s + 4·25-s + 5·27-s − 15·29-s + 4·36-s − 4·39-s − 14·43-s − 4·49-s + 4·51-s − 8·52-s − 13·61-s + 8·64-s + 8·68-s + 3·69-s − 4·75-s − 16·79-s + 81-s + 15·87-s + 6·92-s − 8·100-s − 18·101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 2/3·9-s + 0.577·12-s + 1.10·13-s − 0.970·17-s − 0.625·23-s + 4/5·25-s + 0.962·27-s − 2.78·29-s + 2/3·36-s − 0.640·39-s − 2.13·43-s − 4/7·49-s + 0.560·51-s − 1.10·52-s − 1.66·61-s + 64-s + 0.970·68-s + 0.361·69-s − 0.461·75-s − 1.80·79-s + 1/9·81-s + 1.60·87-s + 0.625·92-s − 4/5·100-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25857 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 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$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−10.39052974902671236333716351154, −9.860872231811332124878372759249, −9.159803160825590712836193964779, −8.793177748511285078905546384537, −8.489869059200689088027120193172, −7.73025563129902308880030400505, −7.01753953546284481017713180157, −6.30397441367977548980476257448, −5.87730420333049805900349435124, −5.17785568754491725077738816180, −4.63053266484768783900441840292, −3.88176273977677092503013380740, −3.18836521892796044243753724990, −1.82778856778948458282652372913, 0,
1.82778856778948458282652372913, 3.18836521892796044243753724990, 3.88176273977677092503013380740, 4.63053266484768783900441840292, 5.17785568754491725077738816180, 5.87730420333049805900349435124, 6.30397441367977548980476257448, 7.01753953546284481017713180157, 7.73025563129902308880030400505, 8.489869059200689088027120193172, 8.793177748511285078905546384537, 9.159803160825590712836193964779, 9.860872231811332124878372759249, 10.39052974902671236333716351154
