L(s) = 1 | + (0.984 − 0.175i)2-s + (0.977 − 0.210i)3-s + (0.938 − 0.345i)4-s + (0.713 + 0.700i)5-s + (0.925 − 0.378i)6-s + (0.329 + 0.944i)7-s + (0.863 − 0.505i)8-s + (0.911 − 0.411i)9-s + (0.825 + 0.564i)10-s + (−0.880 − 0.474i)11-s + (0.844 − 0.535i)12-s + (0.295 − 0.955i)13-s + (0.489 + 0.871i)14-s + (0.844 + 0.535i)15-s + (0.760 − 0.648i)16-s + (−0.520 + 0.854i)17-s + ⋯ |
L(s) = 1 | + (0.984 − 0.175i)2-s + (0.977 − 0.210i)3-s + (0.938 − 0.345i)4-s + (0.713 + 0.700i)5-s + (0.925 − 0.378i)6-s + (0.329 + 0.944i)7-s + (0.863 − 0.505i)8-s + (0.911 − 0.411i)9-s + (0.825 + 0.564i)10-s + (−0.880 − 0.474i)11-s + (0.844 − 0.535i)12-s + (0.295 − 0.955i)13-s + (0.489 + 0.871i)14-s + (0.844 + 0.535i)15-s + (0.760 − 0.648i)16-s + (−0.520 + 0.854i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.519892366 - 0.1517457671i\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.519892366 - 0.1517457671i\) |
\(L(1)\) |
\(\approx\) |
\(2.949520239 - 0.1246614375i\) |
\(L(1)\) |
\(\approx\) |
\(2.949520239 - 0.1246614375i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 179 | \( 1 \) |
good | 2 | \( 1 + (0.984 - 0.175i)T \) |
| 3 | \( 1 + (0.977 - 0.210i)T \) |
| 5 | \( 1 + (0.713 + 0.700i)T \) |
| 7 | \( 1 + (0.329 + 0.944i)T \) |
| 11 | \( 1 + (-0.880 - 0.474i)T \) |
| 13 | \( 1 + (0.295 - 0.955i)T \) |
| 17 | \( 1 + (-0.520 + 0.854i)T \) |
| 19 | \( 1 + (-0.994 + 0.105i)T \) |
| 23 | \( 1 + (0.261 + 0.965i)T \) |
| 29 | \( 1 + (-0.394 - 0.918i)T \) |
| 31 | \( 1 + (-0.0529 + 0.998i)T \) |
| 37 | \( 1 + (0.394 - 0.918i)T \) |
| 41 | \( 1 + (-0.607 - 0.794i)T \) |
| 43 | \( 1 + (0.0176 - 0.999i)T \) |
| 47 | \( 1 + (0.997 - 0.0705i)T \) |
| 53 | \( 1 + (-0.804 - 0.593i)T \) |
| 59 | \( 1 + (-0.999 - 0.0352i)T \) |
| 61 | \( 1 + (0.990 - 0.140i)T \) |
| 67 | \( 1 + (-0.863 - 0.505i)T \) |
| 71 | \( 1 + (-0.550 + 0.835i)T \) |
| 73 | \( 1 + (0.896 + 0.442i)T \) |
| 79 | \( 1 + (-0.362 - 0.932i)T \) |
| 83 | \( 1 + (-0.458 + 0.888i)T \) |
| 89 | \( 1 + (-0.984 - 0.175i)T \) |
| 97 | \( 1 + (0.737 + 0.675i)T \) |
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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
−26.725867257261710592913858633231, −25.93031687794669572208251621492, −25.18870386757140402786876318016, −24.14244142194580779472783677396, −23.581425326370218625996457700, −22.15062453126828932019561753142, −21.0808407860040055990519738309, −20.64463421086228326974330654512, −19.94934899919073258417980037222, −18.43043171848081014247999654919, −16.94220746759505066747959429841, −16.234079357868644049061644576562, −15.05466811049611860456465072513, −14.10331939754720256172161734613, −13.37898396339313865996932613008, −12.730121648953236900864854842948, −11.07534251051810419832126772988, −10.010788982680045199300958629250, −8.717568612114917028236293992926, −7.58932457402364671996432418507, −6.51170413046829814534899778724, −4.76824802515833921399628251375, −4.35064539333480664043181914455, −2.69552291758895121581060238777, −1.63979508053398282339104987273,
1.85715012804740420911251925801, 2.6292999263190586830209752110, 3.65750110330896216278676868242, 5.36869934634642760445521328398, 6.23595068125266511245475002325, 7.57955780061877391670447376727, 8.726239666819538098589708415715, 10.19938122190387069828812596048, 11.02642499065187219479103261865, 12.59648936691693605333319125986, 13.26536662659155551025853916711, 14.18243357578845423001112185573, 15.223817795466343827838719757205, 15.54171028037104059315316458168, 17.52960485373791981738364366787, 18.65803821826544196700358181131, 19.37224548312150986479260831854, 20.68613843628545785131280633913, 21.39483257146648139406978803460, 21.97470899642150359551316658124, 23.31995377261451402813240499078, 24.27909320489602823458663218591, 25.31907116947170446660548897370, 25.61909200792548977805903825881, 26.90012802497720734922962771395