L(s) = 1 | + (0.991 − 0.130i)2-s + (−0.793 − 0.608i)3-s + (0.965 − 0.258i)4-s + (−0.442 + 0.896i)5-s + (−0.866 − 0.5i)6-s + (0.997 + 0.0654i)7-s + (0.923 − 0.382i)8-s + (0.258 + 0.965i)9-s + (−0.321 + 0.946i)10-s + (0.130 − 0.991i)11-s + (−0.923 − 0.382i)12-s + (0.896 + 0.442i)13-s + (0.997 − 0.0654i)14-s + (0.896 − 0.442i)15-s + (0.866 − 0.5i)16-s + (−0.0654 − 0.997i)17-s + ⋯ |
L(s) = 1 | + (0.991 − 0.130i)2-s + (−0.793 − 0.608i)3-s + (0.965 − 0.258i)4-s + (−0.442 + 0.896i)5-s + (−0.866 − 0.5i)6-s + (0.997 + 0.0654i)7-s + (0.923 − 0.382i)8-s + (0.258 + 0.965i)9-s + (−0.321 + 0.946i)10-s + (0.130 − 0.991i)11-s + (−0.923 − 0.382i)12-s + (0.896 + 0.442i)13-s + (0.997 − 0.0654i)14-s + (0.896 − 0.442i)15-s + (0.866 − 0.5i)16-s + (−0.0654 − 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.660173935 - 0.5983128454i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.660173935 - 0.5983128454i\) |
\(L(1)\) |
\(\approx\) |
\(1.715301399 - 0.2754629874i\) |
\(L(1)\) |
\(\approx\) |
\(1.715301399 - 0.2754629874i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (0.991 - 0.130i)T \) |
| 3 | \( 1 + (-0.793 - 0.608i)T \) |
| 5 | \( 1 + (-0.442 + 0.896i)T \) |
| 7 | \( 1 + (0.997 + 0.0654i)T \) |
| 11 | \( 1 + (0.130 - 0.991i)T \) |
| 13 | \( 1 + (0.896 + 0.442i)T \) |
| 17 | \( 1 + (-0.0654 - 0.997i)T \) |
| 19 | \( 1 + (0.555 + 0.831i)T \) |
| 23 | \( 1 + (0.659 - 0.751i)T \) |
| 29 | \( 1 + (0.321 + 0.946i)T \) |
| 31 | \( 1 + (0.608 - 0.793i)T \) |
| 37 | \( 1 + (-0.751 + 0.659i)T \) |
| 41 | \( 1 + (-0.946 + 0.321i)T \) |
| 43 | \( 1 + (-0.258 + 0.965i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.130 + 0.991i)T \) |
| 59 | \( 1 + (-0.659 - 0.751i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.831 + 0.555i)T \) |
| 71 | \( 1 + (-0.946 - 0.321i)T \) |
| 73 | \( 1 + (0.965 + 0.258i)T \) |
| 79 | \( 1 + (-0.382 - 0.923i)T \) |
| 83 | \( 1 + (-0.997 + 0.0654i)T \) |
| 89 | \( 1 + (-0.923 + 0.382i)T \) |
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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
−30.23906846739486605228726152048, −28.586077076034468925490328622491, −28.18499647836333601834227860886, −26.98738639715534558334909942312, −25.50474018372996068054867379082, −24.28971904887403411617525520751, −23.52322258756008650544264480991, −22.77726382266426104906192259160, −21.413260869671714270255454111352, −20.806819555134418660500334056054, −19.846239460346781793254296117897, −17.6681858733020431817443245051, −16.98169976913120296348969367250, −15.573089462510616790764427305156, −15.21477781399986511428058621731, −13.52685530038969535189854861658, −12.30081342524404502734421375787, −11.53909899054440834353069446562, −10.45188999050064176010284843727, −8.61982553817530700624103926747, −7.175031034133373743303887798216, −5.578036636907507329604124530868, −4.7496901837817633343815208396, −3.801856366683497172314564596053, −1.38573551985341876563404576205,
1.31726460654838948876079779794, 3.00919285745367681326631864350, 4.59985457205475280426122116507, 5.90078101423941759712343368812, 6.89662639081944629705501919407, 8.06400208031701170794394404714, 10.64880752539872860225455218614, 11.34052822391748615873870075301, 12.02582346469062630966600178031, 13.64375079449274316430827033954, 14.29513336463265940040601807551, 15.70905473124973050936246948630, 16.72294624537224530748863473480, 18.36525405940407583065557566330, 18.9192388697296074896440464902, 20.501429905336788263193456363547, 21.644315792306490156586259511, 22.582593144700852957803019486926, 23.39517846253035123551070901086, 24.2411962555999811599228220307, 25.129689288553543223186274242884, 26.78007798361434334050728786016, 27.87297335396622333064351664010, 29.153539821032255873102116683693, 29.868930839519437099822351863943