| L(s) = 1 | − 2·2-s − 4·5-s + 8·8-s − 3·9-s + 8·10-s + 4·13-s − 16·16-s + 20·17-s + 6·18-s − 38·25-s − 8·26-s − 52·29-s − 40·34-s + 52·37-s − 32·40-s + 116·41-s + 12·45-s + 50·49-s + 76·50-s − 148·53-s + 104·58-s + 52·61-s + 64·64-s − 16·65-s − 24·72-s − 92·73-s − 104·74-s + ⋯ |
| L(s) = 1 | − 2-s − 4/5·5-s + 8-s − 1/3·9-s + 4/5·10-s + 4/13·13-s − 16-s + 1.17·17-s + 1/3·18-s − 1.51·25-s − 0.307·26-s − 1.79·29-s − 1.17·34-s + 1.40·37-s − 4/5·40-s + 2.82·41-s + 4/15·45-s + 1.02·49-s + 1.51·50-s − 2.79·53-s + 1.79·58-s + 0.852·61-s + 64-s − 0.246·65-s − 1/3·72-s − 1.26·73-s − 1.40·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3147895163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3147895163\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1346 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 382 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1150 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1390 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 11426 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.21368534244247546625260313645, −19.63454363885461930623749185310, −18.88346699411507880536372215953, −18.74904406917276109914630132479, −17.62690934090527446578915444830, −17.33981884772548023069555570954, −16.24609613826692652410721547212, −16.10483807261337693382380385118, −14.93792911843039893260472154080, −14.27703873788863352896282392074, −13.35787087310893432453946414938, −12.51798678506732196418498026374, −11.45876558534976765996277661143, −10.93409908939634533731498037515, −9.751538032754624187969349089619, −9.174383214369402603361117159153, −7.82005355956976890465359257184, −7.71002150047828499977504048565, −5.81330292801615233284569326855, −4.04688556322157586076116458372,
4.04688556322157586076116458372, 5.81330292801615233284569326855, 7.71002150047828499977504048565, 7.82005355956976890465359257184, 9.174383214369402603361117159153, 9.751538032754624187969349089619, 10.93409908939634533731498037515, 11.45876558534976765996277661143, 12.51798678506732196418498026374, 13.35787087310893432453946414938, 14.27703873788863352896282392074, 14.93792911843039893260472154080, 16.10483807261337693382380385118, 16.24609613826692652410721547212, 17.33981884772548023069555570954, 17.62690934090527446578915444830, 18.74904406917276109914630132479, 18.88346699411507880536372215953, 19.63454363885461930623749185310, 20.21368534244247546625260313645
