| L(s) = 1 | − 3-s − 5-s + 9-s − 11-s − 2·13-s + 15-s + 17-s − 2·27-s − 2·29-s + 33-s + 2·39-s − 45-s − 47-s − 51-s + 55-s + 2·65-s − 4·71-s − 2·73-s − 79-s + 2·81-s − 4·83-s − 85-s + 2·87-s − 2·97-s − 99-s − 103-s + 109-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 9-s − 11-s − 2·13-s + 15-s + 17-s − 2·27-s − 2·29-s + 33-s + 2·39-s − 45-s − 47-s − 51-s + 55-s + 2·65-s − 4·71-s − 2·73-s − 79-s + 2·81-s − 4·83-s − 85-s + 2·87-s − 2·97-s − 99-s − 103-s + 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.001634087243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001634087243\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$ | \( ( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.208004666237471646548035870310, −8.372765247895775922316789846984, −7.86519428439525497819822141311, −7.53005861239001141794397994772, −7.40267394829386834699338797138, −7.24501194664841360452335092537, −6.80739465675325071096302125763, −6.06115396146323053010651760243, −5.70177951799821100173811433407, −5.54640364137133478784855125364, −5.21017622692175336726612236322, −4.60341898679440797292524763379, −4.35521299984307247806561806836, −4.04345882523985575165827446863, −3.44833510552664789025519452950, −2.87194927327832905627592696237, −2.65682932067581481402282493049, −1.69810432532893621843676467960, −1.51996097849080123173746990361, −0.02366696605190162706817059499,
0.02366696605190162706817059499, 1.51996097849080123173746990361, 1.69810432532893621843676467960, 2.65682932067581481402282493049, 2.87194927327832905627592696237, 3.44833510552664789025519452950, 4.04345882523985575165827446863, 4.35521299984307247806561806836, 4.60341898679440797292524763379, 5.21017622692175336726612236322, 5.54640364137133478784855125364, 5.70177951799821100173811433407, 6.06115396146323053010651760243, 6.80739465675325071096302125763, 7.24501194664841360452335092537, 7.40267394829386834699338797138, 7.53005861239001141794397994772, 7.86519428439525497819822141311, 8.372765247895775922316789846984, 9.208004666237471646548035870310
