| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 8-s + 9-s + 3·10-s − 12-s − 3·15-s + 16-s + 18-s + 3·20-s + 4·23-s − 24-s + 2·25-s − 27-s − 3·29-s − 3·30-s + 32-s + 36-s + 3·40-s − 4·43-s + 3·45-s + 4·46-s + 8·47-s − 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.288·12-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 0.670·20-s + 0.834·23-s − 0.204·24-s + 2/5·25-s − 0.192·27-s − 0.557·29-s − 0.547·30-s + 0.176·32-s + 1/6·36-s + 0.474·40-s − 0.609·43-s + 0.447·45-s + 0.589·46-s + 1.16·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.357556641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.357556641\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.579435405027121706038966368284, −7.87964732136458981171296722310, −7.55390797318539052318305715199, −6.88665754949820621923432636713, −6.58267136458703297287343549836, −6.13651235641653135575231796953, −5.66920899134320154172671547146, −5.30656323314660621890095567480, −4.89175285083874979766712135205, −4.32103642002694366214457735860, −3.63755707134748147569787867923, −3.09651479469508154954612687272, −2.24949401150157775136292576884, −1.88820406651521985591508999511, −0.923518829146334868175574911329,
0.923518829146334868175574911329, 1.88820406651521985591508999511, 2.24949401150157775136292576884, 3.09651479469508154954612687272, 3.63755707134748147569787867923, 4.32103642002694366214457735860, 4.89175285083874979766712135205, 5.30656323314660621890095567480, 5.66920899134320154172671547146, 6.13651235641653135575231796953, 6.58267136458703297287343549836, 6.88665754949820621923432636713, 7.55390797318539052318305715199, 7.87964732136458981171296722310, 8.579435405027121706038966368284
