| L(s) = 1 | − 3-s − 4-s + 7-s + 12-s + 2·13-s + 19-s − 21-s + 27-s − 28-s + 4·31-s − 2·37-s − 2·39-s + 43-s + 49-s − 2·52-s − 57-s − 2·61-s + 64-s + 67-s − 2·73-s − 76-s − 2·79-s − 81-s + 84-s + 2·91-s − 4·93-s − 2·97-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 7-s + 12-s + 2·13-s + 19-s − 21-s + 27-s − 28-s + 4·31-s − 2·37-s − 2·39-s + 43-s + 49-s − 2·52-s − 57-s − 2·61-s + 64-s + 67-s − 2·73-s − 76-s − 2·79-s − 81-s + 84-s + 2·91-s − 4·93-s − 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6750063549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6750063549\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−10.30453205798940302719406664046, −10.18176660991191079611971912361, −9.675788111142042728602167909177, −8.972571723690020651103881054325, −8.719161452773775644514352834881, −8.355645739792511022006998320294, −8.220949813341573875702182111727, −7.47628791990059865873293477940, −6.99945707754471641375723702135, −6.45918894114801104038988629846, −5.90453157103191901404104566194, −5.85566278588905642219915718281, −5.08766436698850075897926323509, −4.79094098630065470339421470341, −4.38066645668035350937579982274, −3.91891253518455550192590590877, −3.16347999849495124494618806665, −2.65389537912133398228834367011, −1.41571598164591807696615254989, −1.02971418541924725018879198328,
1.02971418541924725018879198328, 1.41571598164591807696615254989, 2.65389537912133398228834367011, 3.16347999849495124494618806665, 3.91891253518455550192590590877, 4.38066645668035350937579982274, 4.79094098630065470339421470341, 5.08766436698850075897926323509, 5.85566278588905642219915718281, 5.90453157103191901404104566194, 6.45918894114801104038988629846, 6.99945707754471641375723702135, 7.47628791990059865873293477940, 8.220949813341573875702182111727, 8.355645739792511022006998320294, 8.719161452773775644514352834881, 8.972571723690020651103881054325, 9.675788111142042728602167909177, 10.18176660991191079611971912361, 10.30453205798940302719406664046
