L(s) = 1 | + (−0.923 + 0.382i)3-s + (0.382 − 0.923i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)13-s + i·15-s − i·17-s + (−0.382 − 0.923i)19-s + (−0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (0.923 − 0.382i)29-s − 31-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)3-s + (0.382 − 0.923i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)13-s + i·15-s − i·17-s + (−0.382 − 0.923i)19-s + (−0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (0.923 − 0.382i)29-s − 31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8119408731 + 0.03988809729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8119408731 + 0.03988809729i\) |
\(L(1)\) |
\(\approx\) |
\(0.9068907647 + 0.03432106584i\) |
\(L(1)\) |
\(\approx\) |
\(0.9068907647 + 0.03432106584i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.923 + 0.382i)T \) |
| 5 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.923 + 0.382i)T \) |
| 13 | \( 1 + (0.382 + 0.923i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (-0.382 - 0.923i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
| 29 | \( 1 + (0.923 - 0.382i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.382 + 0.923i)T \) |
| 41 | \( 1 + (-0.707 + 0.707i)T \) |
| 43 | \( 1 + (-0.923 - 0.382i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.923 + 0.382i)T \) |
| 59 | \( 1 + (0.382 - 0.923i)T \) |
| 61 | \( 1 + (-0.923 + 0.382i)T \) |
| 67 | \( 1 + (-0.923 + 0.382i)T \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 + (-0.382 - 0.923i)T \) |
| 89 | \( 1 + (-0.707 - 0.707i)T \) |
| 97 | \( 1 - T \) |
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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
−32.65874683918747615263502418071, −30.55132098673157590274788015355, −30.1324081960316766360203071220, −29.20043352005840924797988041409, −27.74346015813290854097809867185, −26.98816485460508106737450721647, −25.52318731369244143599038496171, −24.3399081087293895841568794329, −23.223854699107069746557901689172, −22.33989027011093792415485579118, −21.29169446961871940061828308820, −19.66696933749155809769867340566, −18.354273527506476576938099901225, −17.547230354954286438011313375996, −16.55618455407356057411478250208, −14.813705858993439915919031258048, −13.76001694195046695613877820987, −12.31656825464207934577479889254, −10.92086724369123080865205873707, −10.3495703628812869726471435203, −8.10007068692543716373527700145, −6.74129028435277390892562030394, −5.72560160837288939851634712019, −3.8867998504292872916667177647, −1.608912584339406317133466888601,
1.61395579689045687471854883394, 4.35027534755608764687885233959, 5.31292001496009521144944757391, 6.68440481280154351790128167086, 8.75333982704440655081199628750, 9.699973569552783693041862129061, 11.502895944557074895954851675882, 12.08110206505235517563023358434, 13.69032512900771916702788571802, 15.25891623490278612013348392065, 16.41900206014164688741025131006, 17.38285245533884050424775488682, 18.33640954462948472097894328018, 20.102153599416876182273319762010, 21.331499069182291517990092880287, 21.96304534460574951969261945297, 23.48278054620931789364482285476, 24.37485986558361741277942770280, 25.47246761480209979641954751498, 27.185941123978501335132454576794, 28.01222962343578063349895044473, 28.69991573711017823318648471161, 29.913437081431932851964587680133, 31.30283592035135928158501873299, 32.48925824230909645662251550873