L(s) = 1 | − 1.73·5-s + 7-s + 1.73·11-s + 1.73·17-s − 19-s + 1.99·25-s − 1.73·35-s − 43-s − 1.73·47-s − 2.99·55-s + 61-s − 73-s + 1.73·77-s − 2.99·85-s + 1.73·95-s + 1.73·119-s + ⋯ |
L(s) = 1 | − 1.73·5-s + 7-s + 1.73·11-s + 1.73·17-s − 19-s + 1.99·25-s − 1.73·35-s − 43-s − 1.73·47-s − 2.99·55-s + 61-s − 73-s + 1.73·77-s − 2.99·85-s + 1.73·95-s + 1.73·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8643991909\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8643991909\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 1.73T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + 1.73T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.96089162703883707421831415048, −9.849630036228941713633476646511, −8.657757517247381792240336398397, −8.141526658610207652752195967218, −7.34869839358269667892291447727, −6.41756313055707359915304270591, −4.99522220192829773579785439949, −4.10475826781561474544196933959, −3.39926005890127735724322057915, −1.37889148088187987523378442281,
1.37889148088187987523378442281, 3.39926005890127735724322057915, 4.10475826781561474544196933959, 4.99522220192829773579785439949, 6.41756313055707359915304270591, 7.34869839358269667892291447727, 8.141526658610207652752195967218, 8.657757517247381792240336398397, 9.849630036228941713633476646511, 10.96089162703883707421831415048