| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 15-s + 21-s − 23-s + 2·27-s + 2·29-s + 35-s − 2·41-s − 2·43-s + 45-s + 2·47-s − 61-s + 63-s + 67-s − 69-s + 2·81-s − 2·83-s + 2·87-s + 89-s − 101-s − 103-s + 105-s + 107-s − 109-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 15-s + 21-s − 23-s + 2·27-s + 2·29-s + 35-s − 2·41-s − 2·43-s + 45-s + 2·47-s − 61-s + 63-s + 67-s − 69-s + 2·81-s − 2·83-s + 2·87-s + 89-s − 101-s − 103-s + 105-s + 107-s − 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.696004832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.696004832\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.223406622793722033889358104758, −9.052166789529282406851037071577, −8.620503445444294818241416391040, −8.192972544249787449631457148575, −8.060679129234580297922630404262, −7.71507901523219204175097940948, −6.96076898347597237443055853984, −6.74158016808290239950535409007, −6.46857377785885349065098926576, −5.91489349262696394382811965517, −5.22124208985658920448227472268, −5.18398105297171602371041154256, −4.54091209532366868866437900357, −4.25259763376587021656123422318, −3.64564293790094602755200480004, −3.12998924668221957986943907047, −2.55750065147470240143626632587, −2.26447322081202494696376767729, −1.48584274320279708404662926580, −1.34240989926389993025326318431,
1.34240989926389993025326318431, 1.48584274320279708404662926580, 2.26447322081202494696376767729, 2.55750065147470240143626632587, 3.12998924668221957986943907047, 3.64564293790094602755200480004, 4.25259763376587021656123422318, 4.54091209532366868866437900357, 5.18398105297171602371041154256, 5.22124208985658920448227472268, 5.91489349262696394382811965517, 6.46857377785885349065098926576, 6.74158016808290239950535409007, 6.96076898347597237443055853984, 7.71507901523219204175097940948, 8.060679129234580297922630404262, 8.192972544249787449631457148575, 8.620503445444294818241416391040, 9.052166789529282406851037071577, 9.223406622793722033889358104758
