L(s) = 1 | + (0.647 − 0.762i)2-s + (0.796 + 0.605i)3-s + (−0.161 − 0.986i)4-s + (0.468 + 0.883i)5-s + (0.976 − 0.214i)6-s + (−0.994 − 0.108i)7-s + (−0.856 − 0.515i)8-s + (0.267 + 0.963i)9-s + (0.976 + 0.214i)10-s + (−0.947 − 0.319i)11-s + (0.468 − 0.883i)12-s + (0.267 − 0.963i)13-s + (−0.725 + 0.687i)14-s + (−0.161 + 0.986i)15-s + (−0.947 + 0.319i)16-s + (−0.994 + 0.108i)17-s + ⋯ |
L(s) = 1 | + (0.647 − 0.762i)2-s + (0.796 + 0.605i)3-s + (−0.161 − 0.986i)4-s + (0.468 + 0.883i)5-s + (0.976 − 0.214i)6-s + (−0.994 − 0.108i)7-s + (−0.856 − 0.515i)8-s + (0.267 + 0.963i)9-s + (0.976 + 0.214i)10-s + (−0.947 − 0.319i)11-s + (0.468 − 0.883i)12-s + (0.267 − 0.963i)13-s + (−0.725 + 0.687i)14-s + (−0.161 + 0.986i)15-s + (−0.947 + 0.319i)16-s + (−0.994 + 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.352081145 - 0.3088722796i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352081145 - 0.3088722796i\) |
\(L(1)\) |
\(\approx\) |
\(1.475107935 - 0.2846704437i\) |
\(L(1)\) |
\(\approx\) |
\(1.475107935 - 0.2846704437i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (0.647 - 0.762i)T \) |
| 3 | \( 1 + (0.796 + 0.605i)T \) |
| 5 | \( 1 + (0.468 + 0.883i)T \) |
| 7 | \( 1 + (-0.994 - 0.108i)T \) |
| 11 | \( 1 + (-0.947 - 0.319i)T \) |
| 13 | \( 1 + (0.267 - 0.963i)T \) |
| 17 | \( 1 + (-0.994 + 0.108i)T \) |
| 19 | \( 1 + (0.0541 - 0.998i)T \) |
| 23 | \( 1 + (0.907 - 0.419i)T \) |
| 29 | \( 1 + (0.647 + 0.762i)T \) |
| 31 | \( 1 + (0.0541 + 0.998i)T \) |
| 37 | \( 1 + (-0.856 + 0.515i)T \) |
| 41 | \( 1 + (0.907 + 0.419i)T \) |
| 43 | \( 1 + (-0.947 + 0.319i)T \) |
| 47 | \( 1 + (0.468 - 0.883i)T \) |
| 53 | \( 1 + (0.976 - 0.214i)T \) |
| 61 | \( 1 + (0.647 - 0.762i)T \) |
| 67 | \( 1 + (-0.856 - 0.515i)T \) |
| 71 | \( 1 + (0.468 - 0.883i)T \) |
| 73 | \( 1 + (-0.725 + 0.687i)T \) |
| 79 | \( 1 + (0.796 - 0.605i)T \) |
| 83 | \( 1 + (-0.370 - 0.928i)T \) |
| 89 | \( 1 + (0.647 + 0.762i)T \) |
| 97 | \( 1 + (-0.725 - 0.687i)T \) |
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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
−32.69345005153809846041889790901, −31.541804126488735149027671954624, −31.21372188452783818465297325813, −29.532974793881362439590837879546, −28.73057109303876021244753388915, −26.62096856705977238963917466441, −25.75972919328933344123363181127, −24.93992682583606683240276216714, −23.988464169588034809987720692093, −22.94227880711631780354973212744, −21.32870445366448086219317142202, −20.52928382895398957238213195409, −19.01836006450341340073299258260, −17.671536681399613686641449783339, −16.33394019019890013401007110120, −15.33751942605366575251734749442, −13.759218625319049938499871222121, −13.1421064404301538753584134258, −12.18805199953256173286917142128, −9.49807509706427674197115751717, −8.504041034434956002245163189646, −7.09220124168059809875866963020, −5.8551861889986142952376880935, −4.12425293307655424069396832799, −2.42463136647401804823374474297,
2.63366714435129881037046172743, 3.30023432217150018964227261077, 5.11682617858304909781233052056, 6.74452191194648579853622609833, 8.917257850009393669946541522574, 10.26467583843641850763805368356, 10.80706496073625320257156053324, 13.03601882698775239490977695721, 13.63440298893491622912284395721, 15.036122080129688210124609026575, 15.82026868488690968847819077215, 18.080411473744379980217432670078, 19.25577853270634847594987216494, 20.152911238289662921991201335319, 21.402560996889319902677447694498, 22.17997588989722272808174602398, 23.17112685731674955762677782801, 24.893782975404639437684340916442, 26.063452695548884355808923294166, 26.94434276915863630921665576255, 28.46623758493588866434912653755, 29.46400675125528959280723193666, 30.56713868841916427476706983898, 31.44458278177787087367217760491, 32.65540145451218327720258040518