| L(s) = 1 | − 5-s + 2·7-s − 9-s + 2·11-s − 2·13-s + 19-s − 23-s − 2·35-s − 2·37-s + 41-s + 45-s + 2·47-s + 49-s + 53-s − 2·55-s + 2·59-s − 2·63-s + 2·65-s + 4·77-s + 89-s − 4·91-s − 95-s − 2·99-s + 2·103-s + 115-s + 2·117-s + 121-s + ⋯ |
| L(s) = 1 | − 5-s + 2·7-s − 9-s + 2·11-s − 2·13-s + 19-s − 23-s − 2·35-s − 2·37-s + 41-s + 45-s + 2·47-s + 49-s + 53-s − 2·55-s + 2·59-s − 2·63-s + 2·65-s + 4·77-s + 89-s − 4·91-s − 95-s − 2·99-s + 2·103-s + 115-s + 2·117-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.314099306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314099306\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.886269989091098314224352124940, −8.719707224417906527857920457540, −8.314671073311699696086634380387, −7.965064455222957655769838325575, −7.55062751389045490258270488112, −7.36760313654601233156382907999, −6.97852297093724148939125358379, −6.61962447106984075551930490720, −5.78837685987911999117401301561, −5.73887332879638268222262183474, −5.06798101386282369704926604996, −4.93924852574024289387332294438, −4.38003906219632290771811946169, −4.00222382936215213990925273132, −3.71668784814152797422293806826, −3.14876284602560299187250097473, −2.37953556364473894614404126086, −2.11394386211900300612107285870, −1.48731107266169626390219365833, −0.74695730338733601657553426168,
0.74695730338733601657553426168, 1.48731107266169626390219365833, 2.11394386211900300612107285870, 2.37953556364473894614404126086, 3.14876284602560299187250097473, 3.71668784814152797422293806826, 4.00222382936215213990925273132, 4.38003906219632290771811946169, 4.93924852574024289387332294438, 5.06798101386282369704926604996, 5.73887332879638268222262183474, 5.78837685987911999117401301561, 6.61962447106984075551930490720, 6.97852297093724148939125358379, 7.36760313654601233156382907999, 7.55062751389045490258270488112, 7.965064455222957655769838325575, 8.314671073311699696086634380387, 8.719707224417906527857920457540, 8.886269989091098314224352124940
