| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s + 7-s − 8-s + 10-s − 2·13-s + 14-s + 15-s − 16-s + 17-s + 2·19-s + 21-s + 23-s − 24-s + 25-s − 2·26-s − 27-s + 30-s + 34-s + 35-s + 2·38-s − 2·39-s − 40-s + 42-s − 4·43-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s + 7-s − 8-s + 10-s − 2·13-s + 14-s + 15-s − 16-s + 17-s + 2·19-s + 21-s + 23-s − 24-s + 25-s − 2·26-s − 27-s + 30-s + 34-s + 35-s + 2·38-s − 2·39-s − 40-s + 42-s − 4·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.158116730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.158116730\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 23 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.559903733794370035359536627609, −9.350894455665634825915813175140, −8.777503927959084304691888426820, −8.394637039021529159142520994155, −7.918156656559265513221592671703, −7.87701851748890129326974159971, −7.11487224481875368759826745235, −6.77372900700427541987131161141, −6.58921217996425815774353985199, −5.54491691264746875952153422219, −5.38041955204161873916832420171, −5.19062005555007256658895226144, −4.94005786471534710057054977478, −4.36923170371027214131957889596, −3.53244402960607043832521683221, −3.33005458699690209835047358318, −2.88577167346002804629276132508, −2.39283543949854310280213563058, −1.89178482420119405851070783966, −1.15378852336528667087010971805,
1.15378852336528667087010971805, 1.89178482420119405851070783966, 2.39283543949854310280213563058, 2.88577167346002804629276132508, 3.33005458699690209835047358318, 3.53244402960607043832521683221, 4.36923170371027214131957889596, 4.94005786471534710057054977478, 5.19062005555007256658895226144, 5.38041955204161873916832420171, 5.54491691264746875952153422219, 6.58921217996425815774353985199, 6.77372900700427541987131161141, 7.11487224481875368759826745235, 7.87701851748890129326974159971, 7.918156656559265513221592671703, 8.394637039021529159142520994155, 8.777503927959084304691888426820, 9.350894455665634825915813175140, 9.559903733794370035359536627609
