| L(s) = 1 | − 3·3-s + 6·9-s − 6·11-s + 4·13-s − 12·23-s − 9·27-s + 18·33-s − 16·37-s − 12·39-s + 12·47-s + 14·49-s − 24·59-s + 16·61-s + 36·69-s + 12·71-s − 2·73-s + 9·81-s + 18·83-s − 20·97-s − 36·99-s − 6·107-s − 16·109-s + 48·111-s + 24·117-s + 5·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 1.80·11-s + 1.10·13-s − 2.50·23-s − 1.73·27-s + 3.13·33-s − 2.63·37-s − 1.92·39-s + 1.75·47-s + 2·49-s − 3.12·59-s + 2.04·61-s + 4.33·69-s + 1.42·71-s − 0.234·73-s + 81-s + 1.97·83-s − 2.03·97-s − 3.61·99-s − 0.580·107-s − 1.53·109-s + 4.55·111-s + 2.21·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3766121600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3766121600\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 151 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.24241117420089599362030237578, −9.735622760464094679804352445585, −9.160218818393113271881260638511, −8.764439734700187667739256783256, −8.082374327393994916182205016028, −7.960837988361317902228831190964, −7.48843416880746156683661506366, −6.89091377387839359008903746689, −6.59947152126265751554091188917, −6.04781706637437252987046953342, −5.68392438476744828821678133760, −5.38130014074993360084710856524, −5.10645051458956556813116411809, −4.37457029886652802141706842065, −3.93615750866165581470282315462, −3.56055383961580793173395611144, −2.58519152879164453411555551623, −2.06033238712800569075589944275, −1.29936585137234847375855601110, −0.30969000246082014815069030507,
0.30969000246082014815069030507, 1.29936585137234847375855601110, 2.06033238712800569075589944275, 2.58519152879164453411555551623, 3.56055383961580793173395611144, 3.93615750866165581470282315462, 4.37457029886652802141706842065, 5.10645051458956556813116411809, 5.38130014074993360084710856524, 5.68392438476744828821678133760, 6.04781706637437252987046953342, 6.59947152126265751554091188917, 6.89091377387839359008903746689, 7.48843416880746156683661506366, 7.960837988361317902228831190964, 8.082374327393994916182205016028, 8.764439734700187667739256783256, 9.160218818393113271881260638511, 9.735622760464094679804352445585, 10.24241117420089599362030237578
