L(s) = 1 | + (−0.835 + 0.549i)2-s + (0.396 − 0.918i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)7-s + (0.173 + 0.984i)8-s + (0.173 − 0.984i)10-s + (−0.286 + 0.957i)11-s + (−0.0581 + 0.998i)13-s + (0.893 − 0.448i)14-s + (−0.686 − 0.727i)16-s + (−0.939 + 0.342i)17-s + (−0.939 − 0.342i)19-s + (0.396 + 0.918i)20-s + (−0.286 − 0.957i)22-s + (−0.993 + 0.116i)23-s + ⋯ |
L(s) = 1 | + (−0.835 + 0.549i)2-s + (0.396 − 0.918i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)7-s + (0.173 + 0.984i)8-s + (0.173 − 0.984i)10-s + (−0.286 + 0.957i)11-s + (−0.0581 + 0.998i)13-s + (0.893 − 0.448i)14-s + (−0.686 − 0.727i)16-s + (−0.939 + 0.342i)17-s + (−0.939 − 0.342i)19-s + (0.396 + 0.918i)20-s + (−0.286 − 0.957i)22-s + (−0.993 + 0.116i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06798210101 + 0.3138403109i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06798210101 + 0.3138403109i\) |
\(L(1)\) |
\(\approx\) |
\(0.4136351502 + 0.2493305003i\) |
\(L(1)\) |
\(\approx\) |
\(0.4136351502 + 0.2493305003i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.835 + 0.549i)T \) |
| 5 | \( 1 + (-0.686 + 0.727i)T \) |
| 7 | \( 1 + (-0.993 - 0.116i)T \) |
| 11 | \( 1 + (-0.286 + 0.957i)T \) |
| 13 | \( 1 + (-0.0581 + 0.998i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.993 + 0.116i)T \) |
| 29 | \( 1 + (0.893 + 0.448i)T \) |
| 31 | \( 1 + (0.597 + 0.802i)T \) |
| 37 | \( 1 + (0.766 + 0.642i)T \) |
| 41 | \( 1 + (-0.835 - 0.549i)T \) |
| 43 | \( 1 + (0.973 - 0.230i)T \) |
| 47 | \( 1 + (0.597 - 0.802i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.286 - 0.957i)T \) |
| 61 | \( 1 + (0.396 + 0.918i)T \) |
| 67 | \( 1 + (0.893 - 0.448i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (-0.835 + 0.549i)T \) |
| 83 | \( 1 + (-0.835 + 0.549i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (-0.686 - 0.727i)T \) |
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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
−30.29854492391306981043137606634, −29.2502630719260462658675076638, −28.43958367541536774043751615938, −27.4132133949659565882071815065, −26.5500857337169225447282034735, −25.35403345023529149776466980397, −24.3433750957121932440229569275, −22.87099344362361859814835894437, −21.70925587192329047478047485877, −20.44114130776135883411647200883, −19.61930294573911923010275616824, −18.77744259488796196103836789810, −17.394892676922022271251611791288, −16.22190562040864001545873408709, −15.605846270918726122143287215450, −13.277795150670314572416988789084, −12.4725059439973475743504634517, −11.25228778597876118922337283352, −10.01183982565490881773310172282, −8.73701327897348289009940421836, −7.86956036234664347033386932963, −6.19171157033316016139585040025, −4.08978519747750148970540196016, −2.7320469218591978380714065530, −0.44070899055071802399870327433,
2.36489818161407116103996483882, 4.33512038480674171973659034811, 6.44992524422141171980649764297, 7.05836716062445666040158787395, 8.51105997545296256905319200027, 9.84374336551936755084332419450, 10.81235666170689836746666175473, 12.21882298983379598689022427278, 13.96172734148525854883665449484, 15.26400786737337880551995065850, 15.923205463813200622971160639317, 17.22783617113302728305619513109, 18.39893062708316669677148816689, 19.373356241302581837113041610205, 20.07093457004479889811272235056, 21.95295220199889525068398999637, 23.19978999548626934476126258566, 23.863132124367401562982236092995, 25.49434968728473579116015304241, 26.12059654519049118309153476625, 26.97693483761809744499916647591, 28.22953244619765459120191183390, 29.0191305489126129029211332766, 30.33529620166119830088406879770, 31.54831063339679830534870738681