L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.939 + 0.342i)3-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)5-s + (0.766 + 0.642i)6-s + (−0.766 − 1.32i)7-s + 0.999·8-s + (0.766 − 0.642i)9-s − 1.87·10-s + (0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (−0.766 + 1.32i)14-s + (−0.326 + 1.85i)15-s + (−0.5 − 0.866i)16-s + 0.347·17-s + (−0.939 − 0.342i)18-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.939 + 0.342i)3-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)5-s + (0.766 + 0.642i)6-s + (−0.766 − 1.32i)7-s + 0.999·8-s + (0.766 − 0.642i)9-s − 1.87·10-s + (0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (−0.766 + 1.32i)14-s + (−0.326 + 1.85i)15-s + (−0.5 − 0.866i)16-s + 0.347·17-s + (−0.939 − 0.342i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5147218801\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5147218801\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.53T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.911548027305422713524514109507, −9.450684924434015948322092466522, −8.608896895221304053536044766103, −7.37585379946467835636651047575, −6.44779080525321366062332536286, −5.23386789535948569022757166212, −4.50458446429423094879047373854, −3.72377330665349917372713671220, −1.79951200929195876149564536656, −0.66991835851025594725956611114,
1.99331804109701403655199892828, 3.13017099772213926867197817385, 5.20538383011143770870158604305, 5.69359601195021873503094433295, 6.46278532796285625192717192990, 6.90665264903064747012639287727, 7.86921025584560534773685171936, 9.076183722270803004580522147880, 10.02416868392696649899633795196, 10.28075793034175230093771681278