| L(s) = 1 | − 1.73i·2-s − i·3-s − 0.999·4-s − 1.73·6-s − 2.73i·7-s − 1.73i·8-s − 9-s + 4.73·11-s + 0.999i·12-s − 0.732i·13-s − 4.73·14-s − 5·16-s + 1.73i·18-s − 19-s − 2.73·21-s − 8.19i·22-s + ⋯ |
| L(s) = 1 | − 1.22i·2-s − 0.577i·3-s − 0.499·4-s − 0.707·6-s − 1.03i·7-s − 0.612i·8-s − 0.333·9-s + 1.42·11-s + 0.288i·12-s − 0.203i·13-s − 1.26·14-s − 1.25·16-s + 0.408i·18-s − 0.229·19-s − 0.596·21-s − 1.74i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.681800650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681800650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 7 | \( 1 + 2.73iT - 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 0.732iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 6.19iT - 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + 4.19iT - 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 9.46iT - 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 - 16.9iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 6.19iT - 97T^{2} \) |
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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
−9.306653885206434229879097741082, −8.446243127903146220558604137285, −7.28413680150958422239649983214, −6.81468346204564592302521781382, −5.87886160778175022649779287555, −4.31325486800042197658098992426, −3.84110742579424141279933260404, −2.70614984713408326198989218159, −1.58816059526215800369514723902, −0.69572251448223840059976131504,
1.86742139228939138811610787981, 3.18637560587312674456491414749, 4.36962830412626668665455935424, 5.18134402907165270604921705342, 6.12182128120132405028418973598, 6.47261347679879916694530541751, 7.59189307104467560316044389735, 8.369352885101763285156165379661, 9.166012906438243983710981132370, 9.504540385799054678589774400983
