L(s) = 1 | + (−0.939 + 1.62i)7-s + (0.673 + 1.85i)13-s + (0.5 − 0.866i)19-s + (0.939 − 0.342i)25-s + (−1.11 − 0.642i)31-s − 1.28i·37-s + (0.266 + 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (0.439 − 0.524i)67-s + (−0.326 − 0.118i)73-s + (−0.233 + 0.642i)79-s + (−3.64 − 0.642i)91-s + (−1.11 − 1.32i)97-s + (1.70 − 0.984i)103-s + ⋯ |
L(s) = 1 | + (−0.939 + 1.62i)7-s + (0.673 + 1.85i)13-s + (0.5 − 0.866i)19-s + (0.939 − 0.342i)25-s + (−1.11 − 0.642i)31-s − 1.28i·37-s + (0.266 + 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (0.439 − 0.524i)67-s + (−0.326 − 0.118i)73-s + (−0.233 + 0.642i)79-s + (−3.64 − 0.642i)91-s + (−1.11 − 1.32i)97-s + (1.70 − 0.984i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8562109469\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8562109469\) |
\(L(1)\) |
|
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 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.28iT - T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
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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
−11.06352922953166154869721830929, −9.609197801806292359133098823561, −9.172762310488682760614768565863, −8.572535651115436402659249595746, −7.13664619692515108186097279211, −6.36038214156932822595908584551, −5.58901612498828482792561275372, −4.38516822615085000261171675606, −3.11518264521038519215997492338, −2.05313967395021097894259951382,
1.02671385901535761993912402830, 3.20681298209757081057277061821, 3.72134879886700528736049903073, 5.12884487058737318419991300634, 6.14605667666158444645899674368, 7.12547063140059893595785253814, 7.78880347985238086318905752109, 8.807068164156596689130966175959, 10.04150274772410226642764724140, 10.37364632754520356738571892985