| L(s) = 1 | + 3-s + 2·7-s − 9-s + 4·11-s − 4·13-s + 17-s − 3·19-s + 2·21-s + 7·23-s + 11·29-s + 10·31-s + 4·33-s + 3·37-s − 4·39-s − 9·41-s + 9·43-s + 3·49-s + 51-s − 6·53-s − 3·57-s + 2·59-s + 14·61-s − 2·63-s + 24·67-s + 7·69-s + 9·71-s − 17·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s − 1/3·9-s + 1.20·11-s − 1.10·13-s + 0.242·17-s − 0.688·19-s + 0.436·21-s + 1.45·23-s + 2.04·29-s + 1.79·31-s + 0.696·33-s + 0.493·37-s − 0.640·39-s − 1.40·41-s + 1.37·43-s + 3/7·49-s + 0.140·51-s − 0.824·53-s − 0.397·57-s + 0.260·59-s + 1.79·61-s − 0.251·63-s + 2.93·67-s + 0.842·69-s + 1.06·71-s − 1.98·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.379586390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.379586390\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 9 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 102 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24 T + 261 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17 T + 214 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 120 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 184 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.751078221306229559115149321538, −8.557796875491366107573358473057, −8.221641239904386592799671074156, −8.205770535733525264173355539826, −7.48031733045014692840348920895, −6.90994632393524296712583947297, −6.82955784241169488788817066833, −6.58420919518670488803836323396, −5.82275580566750254166763270885, −5.59225864447625533800630998295, −4.82804473167407927343339689817, −4.78516409494978268553488890310, −4.31403547527113414745725085097, −3.90411354483070579537160416063, −3.17269037987444401364522089229, −2.85650336361955474080762333640, −2.46822326680780986771341162871, −1.91496836672215155793115495308, −1.11911502422421208154117074161, −0.75903151500820732143771117379,
0.75903151500820732143771117379, 1.11911502422421208154117074161, 1.91496836672215155793115495308, 2.46822326680780986771341162871, 2.85650336361955474080762333640, 3.17269037987444401364522089229, 3.90411354483070579537160416063, 4.31403547527113414745725085097, 4.78516409494978268553488890310, 4.82804473167407927343339689817, 5.59225864447625533800630998295, 5.82275580566750254166763270885, 6.58420919518670488803836323396, 6.82955784241169488788817066833, 6.90994632393524296712583947297, 7.48031733045014692840348920895, 8.205770535733525264173355539826, 8.221641239904386592799671074156, 8.557796875491366107573358473057, 8.751078221306229559115149321538
