L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + 10-s − 11-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)20-s + (0.5 + 0.866i)22-s + 23-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + 10-s − 11-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)20-s + (0.5 + 0.866i)22-s + 23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1731861736 - 0.4380510773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1731861736 - 0.4380510773i\) |
\(L(1)\) |
\(\approx\) |
\(0.5707492847 - 0.1729354615i\) |
\(L(1)\) |
\(\approx\) |
\(0.5707492847 - 0.1729354615i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.00228315503309361397754236332, −28.63003529431762194146403431096, −27.29992989447889187139220747280, −26.515712996757585624241992378643, −25.62475146203557950622617304875, −24.39869508876239726169160887144, −23.551263700959713101395058545309, −22.99271772663079980925804960130, −21.19206251001443005133338936322, −20.00750805413261850280811574514, −19.16380876021688473483402447464, −17.97282616698046090872956010317, −16.6365301154778564992366542986, −16.268627024369608843898344604916, −15.08875887312005374095422311231, −13.67194533477038955328420984284, −12.83448656502186152301905084661, −11.05381150842799778656207440015, −9.88896605572155668735347697419, −8.6774020710314261146334382238, −7.723358531750207021076452668818, −6.535002557450403002290483829517, −5.09753157102330768031997573950, −3.91373795285137470312594635368, −1.25586006238920775137270620587,
0.26300186164647879883209150835, 2.57454902444139282785457736245, 3.25584682356959233363991786569, 5.1128225659239376604928878020, 6.94349535581063222244872477636, 8.14723827609410907726705191324, 9.33740184349817230579012444655, 10.58417595354474054757725152949, 11.39084289453306125537361127240, 12.59329291259381387452061187711, 13.59645496759416804588355412209, 15.301835065179126544002326318415, 16.06680036638066230645093425222, 17.895670205132920454622593730445, 18.38440996540105037020399430683, 19.37195830253484194471561030649, 20.38063017089029831450835560418, 21.59357804537094595569203226822, 22.46680978889936320127330618634, 23.280948149783554203045088007977, 25.08246981753121429901743269624, 26.01646822791616561491244688949, 26.874721840709850194989140379752, 27.85775951776313323296385540413, 28.84428663386952894681521142245