L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.923 − 0.382i)5-s − 6-s + (−0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (−0.382 − 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.923 + 0.382i)13-s + (−0.923 − 0.382i)14-s + (0.923 − 0.382i)15-s − 16-s + (0.923 + 0.382i)17-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.923 − 0.382i)5-s − 6-s + (−0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (−0.382 − 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.923 + 0.382i)13-s + (−0.923 − 0.382i)14-s + (0.923 − 0.382i)15-s − 16-s + (0.923 + 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01147864501 + 0.6896174230i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01147864501 + 0.6896174230i\) |
\(L(1)\) |
\(\approx\) |
\(0.5619089973 + 0.6236688536i\) |
\(L(1)\) |
\(\approx\) |
\(0.5619089973 + 0.6236688536i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.923 - 0.382i)T \) |
| 7 | \( 1 + (-0.923 + 0.382i)T \) |
| 11 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (0.923 + 0.382i)T \) |
| 17 | \( 1 + (0.923 + 0.382i)T \) |
| 19 | \( 1 + (-0.923 - 0.382i)T \) |
| 23 | \( 1 + (-0.382 + 0.923i)T \) |
| 29 | \( 1 + (-0.382 + 0.923i)T \) |
| 31 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (0.382 + 0.923i)T \) |
| 41 | \( 1 + (0.382 - 0.923i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (-0.382 - 0.923i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.923 + 0.382i)T \) |
| 71 | \( 1 + (0.382 + 0.923i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + (-0.923 - 0.382i)T \) |
| 89 | \( 1 + (-0.707 + 0.707i)T \) |
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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
−29.90010582781692370672477058362, −28.697703983075517933489804523212, −27.9513886970441895662398598355, −26.680361476778250548168062437977, −25.14486638994692398306932809689, −23.80674156449074773245749473684, −23.1252599020562828554822212290, −22.66105622458560607880833010364, −21.26571612376548565977493416222, −19.93194206450467871810847845546, −18.92345329209716775492679226901, −18.46157038067207337214177391742, −16.50280520997351389197614296030, −15.63237939307459377232526190953, −14.04620489709315891930772529282, −13.009090948986143067306060061160, −12.18193020526541552014049652212, −11.00802328151735551886392710949, −10.30745145303655778352894198672, −8.13114796845597112555922807151, −6.668283328060704666763877019745, −5.70847717789904996400793306831, −4.02271608482538590295681293327, −2.78145833969470577747690793465, −0.61470893834242409446203382910,
3.34184514845998857394915016569, 4.3455521713221917493802777037, 5.558776100751779280780473532672, 6.69079768058045742877559198434, 8.15567312024653478815079378386, 9.50009442799559820738050519711, 11.1426613010854796964528914482, 12.26986105456210305556318217392, 13.02668992810146227480077237335, 14.8846907111863502715327417760, 15.73867264818550497826888392092, 16.26349271032566615620016668505, 17.3919015434418162971473258146, 18.842917787858720940820085337285, 20.46497354832516562441609912317, 21.3771750195312373219305090069, 22.50072880251996131735867725685, 23.36679566726832714614594394955, 23.82620941693041473785200948509, 25.63277092542418720573751695532, 26.15670336294888645735316143847, 27.622101372469262019769810036157, 28.302990268340011366893386732557, 29.61161040860200355430708260646, 30.99783970156270141783762290333