L(s) = 1 | + (−0.861 − 0.506i)2-s + (0.779 − 0.626i)3-s + (0.485 + 0.873i)4-s + (−0.262 + 0.964i)5-s + (−0.989 + 0.144i)6-s + (0.885 − 0.464i)7-s + (0.0241 − 0.999i)8-s + (0.215 − 0.976i)9-s + (0.715 − 0.698i)10-s + (0.926 + 0.377i)12-s + (0.885 − 0.464i)13-s + (−0.998 − 0.0483i)14-s + (0.399 + 0.916i)15-s + (−0.527 + 0.849i)16-s + (0.926 − 0.377i)17-s + (−0.681 + 0.732i)18-s + ⋯ |
L(s) = 1 | + (−0.861 − 0.506i)2-s + (0.779 − 0.626i)3-s + (0.485 + 0.873i)4-s + (−0.262 + 0.964i)5-s + (−0.989 + 0.144i)6-s + (0.885 − 0.464i)7-s + (0.0241 − 0.999i)8-s + (0.215 − 0.976i)9-s + (0.715 − 0.698i)10-s + (0.926 + 0.377i)12-s + (0.885 − 0.464i)13-s + (−0.998 − 0.0483i)14-s + (0.399 + 0.916i)15-s + (−0.527 + 0.849i)16-s + (0.926 − 0.377i)17-s + (−0.681 + 0.732i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0710 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0710 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.172534515 - 1.091996063i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.172534515 - 1.091996063i\) |
\(L(1)\) |
\(\approx\) |
\(0.9909406418 - 0.4412525595i\) |
\(L(1)\) |
\(\approx\) |
\(0.9909406418 - 0.4412525595i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.861 - 0.506i)T \) |
| 3 | \( 1 + (0.779 - 0.626i)T \) |
| 5 | \( 1 + (-0.262 + 0.964i)T \) |
| 7 | \( 1 + (0.885 - 0.464i)T \) |
| 13 | \( 1 + (0.885 - 0.464i)T \) |
| 17 | \( 1 + (0.926 - 0.377i)T \) |
| 19 | \( 1 + (0.644 - 0.764i)T \) |
| 23 | \( 1 + (0.958 + 0.285i)T \) |
| 29 | \( 1 + (-0.748 - 0.663i)T \) |
| 31 | \( 1 + (-0.943 - 0.331i)T \) |
| 37 | \( 1 + (0.120 - 0.992i)T \) |
| 41 | \( 1 + (-0.0724 + 0.997i)T \) |
| 43 | \( 1 + (-0.262 + 0.964i)T \) |
| 47 | \( 1 + (0.215 - 0.976i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.926 - 0.377i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.262 - 0.964i)T \) |
| 71 | \( 1 + (-0.607 + 0.794i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.998 + 0.0483i)T \) |
| 83 | \( 1 + (0.485 - 0.873i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (-0.443 + 0.896i)T \) |
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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
−20.64895191998698941614494610780, −20.39961749822910622360918589290, −19.20500699018666991115565444938, −18.796924520632921071275556673132, −17.92963741560663577891666890237, −16.80808365866910860645707423986, −16.49002003702983313073671565285, −15.70434364763990998619728166061, −14.9744676857730739899735301602, −14.37779834716238876451292162772, −13.57260903943053332170352681856, −12.42135529784361455516323076218, −11.47572123310761208507277622784, −10.752321169695489534989535919155, −9.8424730190768420315655527163, −8.94352335530451683567315976310, −8.65241306480576019038340543896, −7.91548488603245420531579326216, −7.19734803693244446847863467889, −5.65320648710477090673286319802, −5.25546762545307134897895782289, −4.2375742406188486788935162263, −3.214160296501390850010754963968, −1.78437395765522077814910603681, −1.32466247259782766110101521380,
0.8170631278399444046509861316, 1.646202087273214633383465218, 2.70416858871779244233245994648, 3.33049706792824883671430212514, 4.120492073252661019401750301218, 5.7391240122876891690885595153, 6.91537653129710558267444141681, 7.49955067506520328520079379426, 7.92186421078074861715637762332, 8.86937202132303588444348372483, 9.65848223403394195813923460097, 10.58158862676876975563215474387, 11.33363433075774974050437827576, 11.7826600009796237456953235839, 13.02521838162502925806505445451, 13.55311383175415341247839706373, 14.5526509672506138899611282253, 15.12617567503783972636692464413, 16.031977704132425441872170542743, 17.0994270029509954538475603846, 17.94209196800781261980966326819, 18.3479046601283707313372355614, 18.91261766825841910670911754928, 19.81326931791354504268248396843, 20.30643804891240459785623195073