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
−26.90012802497720734922962771395, −25.61909200792548977805903825881, −25.31907116947170446660548897370, −24.27909320489602823458663218591, −23.31995377261451402813240499078, −21.97470899642150359551316658124, −21.39483257146648139406978803460, −20.68613843628545785131280633913, −19.37224548312150986479260831854, −18.65803821826544196700358181131, −17.52960485373791981738364366787, −15.54171028037104059315316458168, −15.223817795466343827838719757205, −14.18243357578845423001112185573, −13.26536662659155551025853916711, −12.59648936691693605333319125986, −11.02642499065187219479103261865, −10.19938122190387069828812596048, −8.726239666819538098589708415715, −7.57955780061877391670447376727, −6.23595068125266511245475002325, −5.36869934634642760445521328398, −3.65750110330896216278676868242, −2.6292999263190586830209752110, −1.85715012804740420911251925801,
1.63979508053398282339104987273, 2.69552291758895121581060238777, 4.35064539333480664043181914455, 4.76824802515833921399628251375, 6.51170413046829814534899778724, 7.58932457402364671996432418507, 8.717568612114917028236293992926, 10.010788982680045199300958629250, 11.07534251051810419832126772988, 12.730121648953236900864854842948, 13.37898396339313865996932613008, 14.10331939754720256172161734613, 15.05466811049611860456465072513, 16.234079357868644049061644576562, 16.94220746759505066747959429841, 18.43043171848081014247999654919, 19.94934899919073258417980037222, 20.64463421086228326974330654512, 21.0808407860040055990519738309, 22.15062453126828932019561753142, 23.581425326370218625996457700, 24.14244142194580779472783677396, 25.18870386757140402786876318016, 25.93031687794669572208251621492, 26.725867257261710592913858633231