L(s) = 1 | + (0.939 + 0.342i)3-s + (0.766 + 0.642i)9-s + (0.326 + 0.118i)11-s + (−0.766 + 1.32i)17-s + (−0.939 − 1.62i)19-s + (0.766 − 0.642i)25-s + (0.500 + 0.866i)27-s + (0.266 + 0.223i)33-s + (−1.43 − 1.20i)41-s + (1.43 + 0.524i)43-s + (−0.939 + 0.342i)49-s + (−1.17 + 0.984i)51-s + (−0.326 − 1.85i)57-s + (−1.76 + 0.642i)59-s + (−0.266 − 0.223i)67-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)3-s + (0.766 + 0.642i)9-s + (0.326 + 0.118i)11-s + (−0.766 + 1.32i)17-s + (−0.939 − 1.62i)19-s + (0.766 − 0.642i)25-s + (0.500 + 0.866i)27-s + (0.266 + 0.223i)33-s + (−1.43 − 1.20i)41-s + (1.43 + 0.524i)43-s + (−0.939 + 0.342i)49-s + (−1.17 + 0.984i)51-s + (−0.326 − 1.85i)57-s + (−1.76 + 0.642i)59-s + (−0.266 − 0.223i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.331102533\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.331102533\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
good | 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 11 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 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 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
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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
−10.61236201603248000524133424110, −9.321155889363799463659066853323, −8.842679620603722612074752545963, −8.097081773252442966646105534698, −7.03470844419002393361057611552, −6.26666169654668017338600549621, −4.77382009248142497053692253420, −4.14925594983012218675113337399, −2.95271955397086025546705325804, −1.89445748414108061807889473431,
1.60065576960816549234678278242, 2.80530936422477318736258737986, 3.81160705595061180820198190542, 4.83887828113249312189562565079, 6.20077482773788251645116989162, 6.97299303899456296246809297531, 7.85092010109799200470041525057, 8.644479109612196000086344897302, 9.340682093418770101142682891069, 10.13794684402421276134276754409