| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s − 4·15-s + 4·21-s − 4·23-s + 3·25-s − 4·27-s + 8·29-s + 4·31-s − 4·35-s + 8·37-s + 4·41-s − 12·43-s + 6·45-s − 4·47-s + 3·49-s + 4·53-s + 8·59-s + 16·61-s − 6·63-s + 4·67-s + 8·69-s − 4·71-s + 8·73-s − 6·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 1.03·15-s + 0.872·21-s − 0.834·23-s + 3/5·25-s − 0.769·27-s + 1.48·29-s + 0.718·31-s − 0.676·35-s + 1.31·37-s + 0.624·41-s − 1.82·43-s + 0.894·45-s − 0.583·47-s + 3/7·49-s + 0.549·53-s + 1.04·59-s + 2.04·61-s − 0.755·63-s + 0.488·67-s + 0.963·69-s − 0.474·71-s + 0.936·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.142457422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142457422\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.628915591270526553050448529572, −8.615071378034223709743353068561, −7.902637376282843827630202859548, −7.84907124757866031318234773649, −6.93672243370608077035425387931, −6.91150692810128692480479417832, −6.37244204145764755063101963945, −6.37156518835485607389422716935, −5.70774939376927978870497119638, −5.59165405293377127918187860998, −5.03121919910791507865073994186, −4.70912883705091220668396286966, −4.23849652495195570343495848469, −3.83387771265834082801310775156, −3.19093587872518693354887967796, −2.82523512407135291054938218532, −2.13005030247806279368874212214, −1.85332758713389238361254001689, −0.840062295966529603603512325649, −0.66066861139037611872334236450,
0.66066861139037611872334236450, 0.840062295966529603603512325649, 1.85332758713389238361254001689, 2.13005030247806279368874212214, 2.82523512407135291054938218532, 3.19093587872518693354887967796, 3.83387771265834082801310775156, 4.23849652495195570343495848469, 4.70912883705091220668396286966, 5.03121919910791507865073994186, 5.59165405293377127918187860998, 5.70774939376927978870497119638, 6.37156518835485607389422716935, 6.37244204145764755063101963945, 6.91150692810128692480479417832, 6.93672243370608077035425387931, 7.84907124757866031318234773649, 7.902637376282843827630202859548, 8.615071378034223709743353068561, 8.628915591270526553050448529572
