L(s) = 1 | + (0.608 + 0.793i)2-s + (0.130 − 0.991i)3-s + (−0.258 + 0.965i)4-s + (0.0654 + 0.997i)5-s + (0.866 − 0.5i)6-s + (−0.896 + 0.442i)7-s + (−0.923 + 0.382i)8-s + (−0.965 − 0.258i)9-s + (−0.751 + 0.659i)10-s + (−0.793 − 0.608i)11-s + (0.923 + 0.382i)12-s + (0.997 − 0.0654i)13-s + (−0.896 − 0.442i)14-s + (0.997 + 0.0654i)15-s + (−0.866 − 0.5i)16-s + (−0.442 + 0.896i)17-s + ⋯ |
L(s) = 1 | + (0.608 + 0.793i)2-s + (0.130 − 0.991i)3-s + (−0.258 + 0.965i)4-s + (0.0654 + 0.997i)5-s + (0.866 − 0.5i)6-s + (−0.896 + 0.442i)7-s + (−0.923 + 0.382i)8-s + (−0.965 − 0.258i)9-s + (−0.751 + 0.659i)10-s + (−0.793 − 0.608i)11-s + (0.923 + 0.382i)12-s + (0.997 − 0.0654i)13-s + (−0.896 − 0.442i)14-s + (0.997 + 0.0654i)15-s + (−0.866 − 0.5i)16-s + (−0.442 + 0.896i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08788106392 + 1.072795636i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08788106392 + 1.072795636i\) |
\(L(1)\) |
\(\approx\) |
\(0.8983502998 + 0.5521758387i\) |
\(L(1)\) |
\(\approx\) |
\(0.8983502998 + 0.5521758387i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (0.608 + 0.793i)T \) |
| 3 | \( 1 + (0.130 - 0.991i)T \) |
| 5 | \( 1 + (0.0654 + 0.997i)T \) |
| 7 | \( 1 + (-0.896 + 0.442i)T \) |
| 11 | \( 1 + (-0.793 - 0.608i)T \) |
| 13 | \( 1 + (0.997 - 0.0654i)T \) |
| 17 | \( 1 + (-0.442 + 0.896i)T \) |
| 19 | \( 1 + (-0.831 + 0.555i)T \) |
| 23 | \( 1 + (-0.946 + 0.321i)T \) |
| 29 | \( 1 + (0.751 + 0.659i)T \) |
| 31 | \( 1 + (0.991 + 0.130i)T \) |
| 37 | \( 1 + (0.321 - 0.946i)T \) |
| 41 | \( 1 + (-0.659 + 0.751i)T \) |
| 43 | \( 1 + (0.965 - 0.258i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.793 + 0.608i)T \) |
| 59 | \( 1 + (0.946 + 0.321i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.555 - 0.831i)T \) |
| 71 | \( 1 + (-0.659 - 0.751i)T \) |
| 73 | \( 1 + (-0.258 - 0.965i)T \) |
| 79 | \( 1 + (0.382 + 0.923i)T \) |
| 83 | \( 1 + (0.896 + 0.442i)T \) |
| 89 | \( 1 + (0.923 - 0.382i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.04227691752527239774326067611, −28.46597346033239718024693998324, −27.65394545162715836259128862925, −26.33878643331269606546041208200, −25.25624072398300513775171239328, −23.68761986478560880089334160169, −22.929710381529573124228982010249, −21.85103844041568425521368519020, −20.6467215486789388908445937997, −20.38948795265066843707762305907, −19.1768770171071889852290659785, −17.53869452775498059401595002361, −16.03536708177161433837811454634, −15.52859300226234114069885477588, −13.81693825665176763688628614097, −13.10599653286931721128604749477, −11.81111635228907927478033598337, −10.46347676826918203214310460548, −9.672681204462462365136346990579, −8.56051225391405629198602985684, −6.193863403379895663429843680877, −4.83487408832817830423404917215, −4.02262504735672398908641485852, −2.542858591633289800941445126510, −0.36277053016058231720481824588,
2.52874992670300710733310783892, 3.59646872739423276068856765520, 6.06098093931283590744458305681, 6.263632458083938723632581153186, 7.727348561560650802583160016474, 8.71845373010983654066452379888, 10.71382436175916548509752044803, 12.17685359827671190009932311605, 13.193614687585188562129946783233, 13.97084753172383825465989039547, 15.18483656869270121935801066086, 16.18083660908077678372580592348, 17.6832471701826050672692258001, 18.498367193105551431379548396989, 19.42502030144788876877514245343, 21.1979966502972764481655250523, 22.254612743269546770341258694957, 23.2327073328152954398335925742, 23.88409457990772549538685582863, 25.33668436701449358686869856661, 25.7677228719349824475431558480, 26.666639539041811033795633872915, 28.523330728909820784813292752, 29.67650910751365292636415409016, 30.416418714063817461296172825844