| L(s) = 1 | − 2·3-s − 9-s − 2·11-s − 2·13-s − 2·17-s + 4·19-s − 4·23-s + 6·27-s + 2·29-s + 12·31-s + 4·33-s − 4·37-s + 4·39-s + 12·41-s + 12·43-s − 18·47-s + 4·51-s − 16·53-s − 8·57-s + 12·59-s − 8·61-s + 8·69-s + 12·71-s + 16·73-s − 18·79-s − 4·81-s − 4·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1.15·27-s + 0.371·29-s + 2.15·31-s + 0.696·33-s − 0.657·37-s + 0.640·39-s + 1.87·41-s + 1.82·43-s − 2.62·47-s + 0.560·51-s − 2.19·53-s − 1.05·57-s + 1.56·59-s − 1.02·61-s + 0.963·69-s + 1.42·71-s + 1.87·73-s − 2.02·79-s − 4/9·81-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 173 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 132 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 207 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 193 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.42735595810128330478000457348, −7.18779256961290935423428134968, −6.65692833283134968216736881342, −6.38004131890985047462811067508, −6.10945973186793186291165590726, −5.93790018229497773635797935430, −5.27410718495524675305981198899, −5.26640716718316782132296590146, −4.73988158462532262795917855376, −4.64802319246514863998535508912, −4.03531056985308966130380679726, −3.72332938339235997388163599074, −3.02491710069489775876042114060, −2.85562725795060132773501730464, −2.43593051239561664514041145368, −2.04029021116062404147737343646, −1.18490698918217620593356052274, −0.963746296292110783962708038889, 0, 0,
0.963746296292110783962708038889, 1.18490698918217620593356052274, 2.04029021116062404147737343646, 2.43593051239561664514041145368, 2.85562725795060132773501730464, 3.02491710069489775876042114060, 3.72332938339235997388163599074, 4.03531056985308966130380679726, 4.64802319246514863998535508912, 4.73988158462532262795917855376, 5.26640716718316782132296590146, 5.27410718495524675305981198899, 5.93790018229497773635797935430, 6.10945973186793186291165590726, 6.38004131890985047462811067508, 6.65692833283134968216736881342, 7.18779256961290935423428134968, 7.42735595810128330478000457348
