L(s) = 1 | − 1.73i·5-s + i·7-s + 1.73·11-s − 1.99·25-s − i·31-s + 1.73·35-s − 1.73i·53-s − 2.99i·55-s + 73-s + 1.73i·77-s + 2i·79-s − 1.73·83-s − 97-s − 1.73i·101-s + 2i·103-s + ⋯ |
L(s) = 1 | − 1.73i·5-s + i·7-s + 1.73·11-s − 1.99·25-s − i·31-s + 1.73·35-s − 1.73i·53-s − 2.99i·55-s + 73-s + 1.73i·77-s + 2i·79-s − 1.73·83-s − 97-s − 1.73i·101-s + 2i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.226193708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.226193708\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.73iT - T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.73iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.460297355588481295784478368472, −8.586311973124528958696126120511, −8.259837785756685271928415424213, −6.96554693003475030939259925085, −6.01058269903611076093421220034, −5.36461389378710630803584534790, −4.46875602335939669220137224221, −3.72500822319344091911680340217, −2.15091601406651104804112614109, −1.12668990585797744293638940924,
1.48913112638116237446730664632, 2.87473423681427240939004859718, 3.69020245367746803452221434075, 4.35252477490520141037581438516, 5.85216650966682629769221314303, 6.69667009869626762415287951612, 6.99490072783038497509693286017, 7.79244623976972059789088440157, 8.936986170836669074269077289324, 9.736917408351408541347054304305