| L(s) = 1 | − 2·5-s − 2·7-s + 2·11-s − 2·13-s − 6·17-s + 3·25-s − 16·29-s − 4·31-s + 4·35-s + 8·37-s − 16·41-s + 14·43-s + 4·47-s + 2·49-s − 4·53-s − 4·55-s + 16·59-s + 4·65-s + 8·67-s − 6·73-s − 4·77-s + 8·79-s − 14·83-s + 12·85-s − 12·89-s + 4·91-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.603·11-s − 0.554·13-s − 1.45·17-s + 3/5·25-s − 2.97·29-s − 0.718·31-s + 0.676·35-s + 1.31·37-s − 2.49·41-s + 2.13·43-s + 0.583·47-s + 2/7·49-s − 0.549·53-s − 0.539·55-s + 2.08·59-s + 0.496·65-s + 0.977·67-s − 0.702·73-s − 0.455·77-s + 0.900·79-s − 1.53·83-s + 1.30·85-s − 1.27·89-s + 0.419·91-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9204896980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9204896980\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 202 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 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
−7.86739059511417447780593825325, −7.60017366118396307950429390639, −7.30436264552640192558414271083, −7.00891872598201563558799975826, −6.77102552405242496745151760111, −6.36115567327662924821650293141, −5.87654132925902779339127414677, −5.70158671667094594990176892892, −5.15385892152145313954897201307, −4.91414983680240802868792780267, −4.22913287048697234101026021481, −4.11232525833053061653749322358, −3.75246420633911380817874852486, −3.50025555899855534814454861781, −2.82983682538495676475230197901, −2.57633097418736346941353339379, −1.87555099932477660171282438902, −1.76623024696913363788142330246, −0.74377489692738646437579134690, −0.30564685641399426488259525472,
0.30564685641399426488259525472, 0.74377489692738646437579134690, 1.76623024696913363788142330246, 1.87555099932477660171282438902, 2.57633097418736346941353339379, 2.82983682538495676475230197901, 3.50025555899855534814454861781, 3.75246420633911380817874852486, 4.11232525833053061653749322358, 4.22913287048697234101026021481, 4.91414983680240802868792780267, 5.15385892152145313954897201307, 5.70158671667094594990176892892, 5.87654132925902779339127414677, 6.36115567327662924821650293141, 6.77102552405242496745151760111, 7.00891872598201563558799975826, 7.30436264552640192558414271083, 7.60017366118396307950429390639, 7.86739059511417447780593825325
