L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.955 + 0.294i)3-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + (−0.623 − 0.781i)8-s + (0.826 − 0.563i)9-s + (−0.988 − 0.149i)10-s + (−0.900 − 0.433i)11-s + (0.733 − 0.680i)12-s + (−0.988 + 0.149i)13-s + (0.955 + 0.294i)14-s + (0.0747 − 0.997i)15-s + (0.623 − 0.781i)16-s + (0.365 + 0.930i)17-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.955 + 0.294i)3-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + (−0.623 − 0.781i)8-s + (0.826 − 0.563i)9-s + (−0.988 − 0.149i)10-s + (−0.900 − 0.433i)11-s + (0.733 − 0.680i)12-s + (−0.988 + 0.149i)13-s + (0.955 + 0.294i)14-s + (0.0747 − 0.997i)15-s + (0.623 − 0.781i)16-s + (0.365 + 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1250375501 + 0.1805530239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1250375501 + 0.1805530239i\) |
\(L(1)\) |
\(\approx\) |
\(0.4275115564 + 0.3748081313i\) |
\(L(1)\) |
\(\approx\) |
\(0.4275115564 + 0.3748081313i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 3 | \( 1 + (-0.955 + 0.294i)T \) |
| 5 | \( 1 + (-0.365 + 0.930i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.988 + 0.149i)T \) |
| 17 | \( 1 + (0.365 + 0.930i)T \) |
| 19 | \( 1 + (-0.826 - 0.563i)T \) |
| 23 | \( 1 + (0.0747 + 0.997i)T \) |
| 29 | \( 1 + (-0.955 - 0.294i)T \) |
| 31 | \( 1 + (-0.733 + 0.680i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.222 - 0.974i)T \) |
| 47 | \( 1 + (-0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.988 - 0.149i)T \) |
| 59 | \( 1 + (0.623 - 0.781i)T \) |
| 61 | \( 1 + (0.733 + 0.680i)T \) |
| 67 | \( 1 + (0.826 + 0.563i)T \) |
| 71 | \( 1 + (-0.0747 + 0.997i)T \) |
| 73 | \( 1 + (0.988 - 0.149i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.955 - 0.294i)T \) |
| 89 | \( 1 + (-0.955 + 0.294i)T \) |
| 97 | \( 1 + (-0.900 - 0.433i)T \) |
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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
−33.57871806628068491194341301101, −31.94643472014477988468026335569, −31.251989077111195308091297736, −29.809458603941575022502593357712, −28.76474748285393266686628173529, −28.00405203363356451100227785227, −27.159863332550893205816978778871, −24.77300320420201144499952107375, −23.77679510664625103613556732050, −22.70766515956387291158466422098, −21.46751832391095958745176980367, −20.45053245936592994889796577843, −18.896087428665260191071388336226, −17.9663232599222608218886977102, −16.58175704348090475565194994067, −14.91984141500100053197911641867, −12.872818698552231818394818570267, −12.29861523062645019431849770079, −11.16179887148027960824569002792, −9.631923844781590780236393415990, −7.95562295066321424785749238056, −5.46224488549848915982855278291, −4.66756977274747501083708973541, −2.078374050380864804405822345808, −0.13836761956586586279494808628,
3.8383850762264910680237064742, 5.25453097316806927785063194337, 6.76247294655105446528772989248, 7.80648260500463909647171327985, 10.06599665881085891590828952233, 11.24293473418147683348526763184, 12.95054336231972744487933334669, 14.52793885838560990517512245428, 15.53461791301013505519421119880, 16.88964009332190311468665058748, 17.73192081559039749101038780988, 19.04575664179352193173705494481, 21.344753184180159899802725286912, 22.255206404919201343271479106734, 23.607797341067793153549119689573, 23.85034845937961424691500453278, 25.96085633523626948889151544693, 26.835276573044053013835297554743, 27.65745932255504095658480468503, 29.46190713043536766419962681439, 30.5167949925419022818496158567, 31.94360930159370394327566410156, 33.17471762289274137055758676357, 34.21193183255826171626548445883, 34.49681834576204887027177984933