| L(s) = 1 | + 3·3-s + 4·5-s − 2·7-s + 6·9-s + 5·11-s − 2·13-s + 12·15-s − 6·17-s + 2·19-s − 6·21-s − 6·23-s + 5·25-s + 9·27-s − 2·29-s − 4·31-s + 15·33-s − 8·35-s + 16·37-s − 6·39-s − 41-s + 7·43-s + 24·45-s + 2·47-s + 7·49-s − 18·51-s + 8·53-s + 20·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.78·5-s − 0.755·7-s + 2·9-s + 1.50·11-s − 0.554·13-s + 3.09·15-s − 1.45·17-s + 0.458·19-s − 1.30·21-s − 1.25·23-s + 25-s + 1.73·27-s − 0.371·29-s − 0.718·31-s + 2.61·33-s − 1.35·35-s + 2.63·37-s − 0.960·39-s − 0.156·41-s + 1.06·43-s + 3.57·45-s + 0.291·47-s + 49-s − 2.52·51-s + 1.09·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.701765887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.701765887\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.75517737425482418044524040959, −10.03146618480358117410861531935, −10.00726799784679649735215587144, −9.442396673457230160521912035213, −9.243217227527519171455507767977, −9.048238395078110363851809613977, −8.490267108338694143551235644450, −7.895419256837656918671710299719, −7.30165458041020094032100945604, −7.03043548563874145731304219600, −6.32297896590250960601168403954, −6.07519288500321125832662026857, −5.64174789599990200284470977679, −4.69911344198151850806815910000, −3.97215094517217592544015534420, −3.95124916271460785385898270915, −2.94200606822705141090101637577, −2.29476337896407485069167871132, −2.16571508983059019560353642458, −1.27072654520494929129586247714,
1.27072654520494929129586247714, 2.16571508983059019560353642458, 2.29476337896407485069167871132, 2.94200606822705141090101637577, 3.95124916271460785385898270915, 3.97215094517217592544015534420, 4.69911344198151850806815910000, 5.64174789599990200284470977679, 6.07519288500321125832662026857, 6.32297896590250960601168403954, 7.03043548563874145731304219600, 7.30165458041020094032100945604, 7.895419256837656918671710299719, 8.490267108338694143551235644450, 9.048238395078110363851809613977, 9.243217227527519171455507767977, 9.442396673457230160521912035213, 10.00726799784679649735215587144, 10.03146618480358117410861531935, 10.75517737425482418044524040959
