L(s) = 1 | + (0.326 − 1.85i)5-s + (−0.173 − 0.984i)11-s + (−1.53 − 1.28i)23-s + (−2.37 − 0.866i)25-s + (1.17 + 0.984i)31-s + (−0.173 + 0.300i)37-s + (−0.266 + 0.223i)47-s + (0.173 − 0.984i)49-s + 1.87·53-s − 1.87·55-s + (−0.266 + 1.50i)59-s + (−0.326 + 0.118i)67-s + (−0.939 + 1.62i)71-s + (−0.5 − 0.866i)89-s + (−0.326 − 1.85i)97-s + ⋯ |
L(s) = 1 | + (0.326 − 1.85i)5-s + (−0.173 − 0.984i)11-s + (−1.53 − 1.28i)23-s + (−2.37 − 0.866i)25-s + (1.17 + 0.984i)31-s + (−0.173 + 0.300i)37-s + (−0.266 + 0.223i)47-s + (0.173 − 0.984i)49-s + 1.87·53-s − 1.87·55-s + (−0.266 + 1.50i)59-s + (−0.326 + 0.118i)67-s + (−0.939 + 1.62i)71-s + (−0.5 − 0.866i)89-s + (−0.326 − 1.85i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.134071439\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.134071439\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (0.173 + 0.984i)T \) |
good | 5 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.53 + 1.28i)T + (0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 37 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 - 1.87T + T^{2} \) |
| 59 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
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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
−8.505799665663895311735448510601, −8.160427386596206374520522915139, −6.98301580816172454355222424778, −5.92289183374767119859073488730, −5.58612396980053449657839002494, −4.58935690207823350881702120015, −4.11536485151312673655853637153, −2.82470380799485949380980918995, −1.68604149571420675361612135135, −0.64250097897591725571505006307,
1.90075611597550720504709953844, 2.52360746216439309298021091045, 3.48889301471235693460294192213, 4.22159915822767668140689062853, 5.41547308018899570110719948842, 6.19307836823763087109954106020, 6.71792519265107196812772907398, 7.62700886013464459055215448822, 7.83297267420634851590103257827, 9.232476822043363686313259463844