L(s) = 1 | + (−0.923 − 0.382i)2-s + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)4-s + (0.195 − 0.980i)5-s + i·6-s + (0.195 + 0.980i)7-s + (−0.382 − 0.923i)8-s + (−0.707 + 0.707i)9-s + (−0.555 + 0.831i)10-s + (0.382 + 0.923i)11-s + (0.382 − 0.923i)12-s + (−0.980 − 0.195i)13-s + (0.195 − 0.980i)14-s + (−0.980 + 0.195i)15-s + i·16-s + (−0.980 − 0.195i)17-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (−0.382 − 0.923i)3-s + (0.707 + 0.707i)4-s + (0.195 − 0.980i)5-s + i·6-s + (0.195 + 0.980i)7-s + (−0.382 − 0.923i)8-s + (−0.707 + 0.707i)9-s + (−0.555 + 0.831i)10-s + (0.382 + 0.923i)11-s + (0.382 − 0.923i)12-s + (−0.980 − 0.195i)13-s + (0.195 − 0.980i)14-s + (−0.980 + 0.195i)15-s + i·16-s + (−0.980 − 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5484279878 + 0.1986804735i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5484279878 + 0.1986804735i\) |
\(L(1)\) |
\(\approx\) |
\(0.5664984035 - 0.1356318603i\) |
\(L(1)\) |
\(\approx\) |
\(0.5664984035 - 0.1356318603i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.195 - 0.980i)T \) |
| 7 | \( 1 + (0.195 + 0.980i)T \) |
| 11 | \( 1 + (0.382 + 0.923i)T \) |
| 13 | \( 1 + (-0.980 - 0.195i)T \) |
| 17 | \( 1 + (-0.980 - 0.195i)T \) |
| 19 | \( 1 + (-0.195 + 0.980i)T \) |
| 23 | \( 1 + (0.555 + 0.831i)T \) |
| 29 | \( 1 + (0.555 + 0.831i)T \) |
| 31 | \( 1 + (0.923 - 0.382i)T \) |
| 37 | \( 1 + (0.831 + 0.555i)T \) |
| 41 | \( 1 + (-0.831 + 0.555i)T \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.382 - 0.923i)T \) |
| 59 | \( 1 + (-0.555 + 0.831i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.980 - 0.195i)T \) |
| 71 | \( 1 + (-0.831 - 0.555i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 + (-0.923 + 0.382i)T \) |
| 83 | \( 1 + (-0.195 + 0.980i)T \) |
| 89 | \( 1 + (0.382 + 0.923i)T \) |
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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
−29.40579775996159937974922851161, −28.63492703896029901273745637420, −27.133969246080949993464060668580, −26.77062218281186084059253004203, −26.12824257508922291867983603152, −24.6210871554925777770200127184, −23.50258742815677095135908313337, −22.34927339701372092249538887378, −21.32467796864765197708225809265, −20.00151780840254769600464303694, −19.096835809016994089337424691439, −17.551277160506339527068393432319, −17.13447433758943222565694895090, −15.88638341275151109438981756283, −14.84323029243111221923001193370, −13.947646408294717178733165641424, −11.47053384732146704648029646609, −10.781973429406940398450455490268, −9.9299992930024951799862336596, −8.70459630429188317263034166409, −7.089769761444108534960574190933, −6.19896609681969638423705967268, −4.52320794204955816836664285861, −2.74208514188709039770229416514, −0.37926829059784614000341983268,
1.38690856108856698407691299988, 2.42078340491806252461064175766, 4.92340232135940723667117613902, 6.41809610482027274186585129236, 7.74977043643529361796918525108, 8.76862886355002092940885395905, 9.8625416769169325201402729614, 11.61634910995233067570014713656, 12.24917669321707497327667332544, 13.10959949091901009128673894501, 15.05216490245258965205872904846, 16.47465480022747568302965513936, 17.450547264973118183327324654939, 18.07387512127136836533986624211, 19.35779365699155255368224044176, 20.0978783764473819601604511346, 21.35453537771808255267128907664, 22.50452593343806727292796165160, 24.115291486407840352363102573264, 24.994803630203881311635652114324, 25.38914117822416242664470004454, 27.24187081927452773363089447376, 28.0618631875830571952232013729, 28.870479063067337087251207070007, 29.54056750981098103919675952998