L(s) = 1 | + (−0.939 − 0.342i)2-s + (−0.866 + 0.5i)3-s + (0.766 + 0.642i)4-s + (0.906 + 0.422i)5-s + (0.984 − 0.173i)6-s + (−0.258 + 0.965i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 − 0.707i)10-s + (0.422 − 0.906i)11-s + (−0.984 − 0.173i)12-s + (0.819 + 0.573i)13-s + (0.573 − 0.819i)14-s + (−0.996 + 0.0871i)15-s + (0.173 + 0.984i)16-s + (−0.965 + 0.258i)17-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (−0.866 + 0.5i)3-s + (0.766 + 0.642i)4-s + (0.906 + 0.422i)5-s + (0.984 − 0.173i)6-s + (−0.258 + 0.965i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 − 0.707i)10-s + (0.422 − 0.906i)11-s + (−0.984 − 0.173i)12-s + (0.819 + 0.573i)13-s + (0.573 − 0.819i)14-s + (−0.996 + 0.0871i)15-s + (0.173 + 0.984i)16-s + (−0.965 + 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4088131940 + 0.5944929342i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4088131940 + 0.5944929342i\) |
\(L(1)\) |
\(\approx\) |
\(0.5860461604 + 0.2055288782i\) |
\(L(1)\) |
\(\approx\) |
\(0.5860461604 + 0.2055288782i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.906 + 0.422i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
| 11 | \( 1 + (0.422 - 0.906i)T \) |
| 13 | \( 1 + (0.819 + 0.573i)T \) |
| 17 | \( 1 + (-0.965 + 0.258i)T \) |
| 19 | \( 1 + (-0.342 + 0.939i)T \) |
| 23 | \( 1 + (0.342 + 0.939i)T \) |
| 29 | \( 1 + (-0.906 + 0.422i)T \) |
| 31 | \( 1 + (0.0871 - 0.996i)T \) |
| 37 | \( 1 + (-0.939 + 0.342i)T \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.965 - 0.258i)T \) |
| 47 | \( 1 + (0.573 + 0.819i)T \) |
| 53 | \( 1 + (-0.422 - 0.906i)T \) |
| 59 | \( 1 + (-0.573 + 0.819i)T \) |
| 61 | \( 1 + (0.984 + 0.173i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 79 | \( 1 + (0.984 - 0.173i)T \) |
| 83 | \( 1 + (-0.707 - 0.707i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−30.36376259034764249573332165951, −29.65367059755281371171786773817, −28.56040793714143960227566538849, −28.03521814347594803616570002546, −26.59796941987309680208646862755, −25.44585609359144039037757174497, −24.59625304196595206828309824408, −23.51138326152609032306899857327, −22.42938193554447595077877731057, −20.70619182830550903659272402471, −19.77687291788514692587756366228, −18.28094764124006454937954195281, −17.45344774009839922863001094419, −16.86124315681510871252371173090, −15.609330322819505242133837326020, −13.79366325666443081700476013954, −12.67068671730824718927713680797, −11.014442319974818077017917520069, −10.17889949866940543059863148813, −8.81808543245663019474301778155, −7.124293498468047932264833944285, −6.37514989304405470922525578019, −4.90634637996989307433660649488, −1.89496747052554594074664705706, −0.549264606168342424805172506571,
1.6777125819572448823540567455, 3.49180301974856514454791413234, 5.83066247432272865300861590236, 6.562997859816718284588525819113, 8.7688810320906765678133923069, 9.62418484394379760874340667569, 10.88967448087434980476097969149, 11.67473913471045111861226095596, 13.15526667799569628975644178394, 15.09625990294892670473377574128, 16.269936501638585815840409706414, 17.18962005265630229039013170672, 18.291680941712052287898857307129, 18.99804931833626895840590124043, 20.87186463760591661067316411165, 21.6488766072831364648429764165, 22.374288894233694604066678222690, 24.21027816972620602619511648895, 25.399324691267808426632310743646, 26.35904755729454456570927871821, 27.44981568603207770834068111807, 28.45673915079036273362827038158, 29.16481672288147227077896516673, 30.00176654078064440096863785853, 31.57191029890717237617300544714