| L(s) = 1 | + 3-s + 4-s + 9-s + 12-s + 16-s + 19-s + 25-s + 27-s − 17·29-s + 36-s − 41-s − 9·43-s + 48-s − 13·49-s − 15·53-s + 57-s + 7·59-s − 2·61-s + 64-s − 14·71-s − 8·73-s + 75-s + 76-s + 81-s − 17·87-s + 2·89-s + 100-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1/3·9-s + 0.288·12-s + 1/4·16-s + 0.229·19-s + 1/5·25-s + 0.192·27-s − 3.15·29-s + 1/6·36-s − 0.156·41-s − 1.37·43-s + 0.144·48-s − 1.85·49-s − 2.06·53-s + 0.132·57-s + 0.911·59-s − 0.256·61-s + 1/8·64-s − 1.66·71-s − 0.936·73-s + 0.115·75-s + 0.114·76-s + 1/9·81-s − 1.82·87-s + 0.211·89-s + 1/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 105 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
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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
−7.955331180956713891706450845547, −7.64920986279308508298967757556, −7.26173150870952141562443374552, −6.82030958849751929460368441401, −6.28552390215563837657747208938, −5.86876864151322244734414670823, −5.30554288065168214300977193404, −4.86131601613778483632987359229, −4.24353223882677329089338326355, −3.57821691678704926149686205580, −3.28618503244237244378093076216, −2.67551926576351289013673538094, −1.76517839824886496161682978039, −1.60042173582177710982185638377, 0,
1.60042173582177710982185638377, 1.76517839824886496161682978039, 2.67551926576351289013673538094, 3.28618503244237244378093076216, 3.57821691678704926149686205580, 4.24353223882677329089338326355, 4.86131601613778483632987359229, 5.30554288065168214300977193404, 5.86876864151322244734414670823, 6.28552390215563837657747208938, 6.82030958849751929460368441401, 7.26173150870952141562443374552, 7.64920986279308508298967757556, 7.955331180956713891706450845547
