L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.5 − 0.866i)8-s + (−0.266 − 0.223i)11-s + (0.766 + 0.642i)16-s + (0.766 + 1.32i)17-s + (0.939 − 1.62i)19-s + (0.266 − 0.223i)22-s + (0.173 − 0.984i)25-s + (−0.766 + 0.642i)32-s + (−1.43 + 0.524i)34-s + (1.43 + 1.20i)38-s + (0.326 + 1.85i)41-s + (1.17 + 0.984i)43-s + (0.173 + 0.300i)44-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.5 − 0.866i)8-s + (−0.266 − 0.223i)11-s + (0.766 + 0.642i)16-s + (0.766 + 1.32i)17-s + (0.939 − 1.62i)19-s + (0.266 − 0.223i)22-s + (0.173 − 0.984i)25-s + (−0.766 + 0.642i)32-s + (−1.43 + 0.524i)34-s + (1.43 + 1.20i)38-s + (0.326 + 1.85i)41-s + (1.17 + 0.984i)43-s + (0.173 + 0.300i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9832605556\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9832605556\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 11 | \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \) |
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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
−9.411371225809845181008039888332, −8.502717643764735603027314561612, −7.988894510284683232804516294409, −7.15006838775406302048756979370, −6.37452350295064568055858745735, −5.62662573891338397163603610048, −4.81643388169252996442266453164, −3.93458893933579024584713088547, −2.77222240492862098589866283036, −1.05668458252014961602652953925,
1.09974060097411744207754349852, 2.32491104903267342083480341237, 3.30925062068482923950644190547, 4.06123165961833433305203749392, 5.25111156293994124699886461842, 5.67617980370340602608740618576, 7.36088447650698544628659429614, 7.59838398391012036183469770841, 8.750145679815994650132288469659, 9.365929646680664192323400806801