| L(s) = 1 | + 2·3-s + 3·9-s + 4·11-s + 4·17-s + 8·19-s − 8·25-s + 4·27-s + 8·33-s + 12·41-s + 16·43-s − 12·49-s + 8·51-s + 16·57-s + 24·59-s + 16·67-s − 16·73-s − 16·75-s + 5·81-s + 12·83-s + 4·89-s − 28·97-s + 12·99-s + 8·107-s + 12·113-s − 10·121-s + 24·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.20·11-s + 0.970·17-s + 1.83·19-s − 8/5·25-s + 0.769·27-s + 1.39·33-s + 1.87·41-s + 2.43·43-s − 1.71·49-s + 1.12·51-s + 2.11·57-s + 3.12·59-s + 1.95·67-s − 1.87·73-s − 1.84·75-s + 5/9·81-s + 1.31·83-s + 0.423·89-s − 2.84·97-s + 1.20·99-s + 0.773·107-s + 1.12·113-s − 0.909·121-s + 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.964457321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.964457321\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−9.623023746496957437399267145123, −9.344362040845107118359055715455, −8.898230127423678495459227882582, −8.475374982140391211858934967833, −8.034776072701869713369129177743, −7.63614701002364541388417247504, −7.39560216304176002028261402091, −7.06589728763642699759212514089, −6.39227626325011928837343615527, −6.04534803929674304450213973961, −5.45215002417403430665749076969, −5.27007052034441234264207157605, −4.30824110962678800119848532316, −4.15703626568098124364510977627, −3.48616781407495673743751503909, −3.41643432170277313086471770107, −2.51249062672094151589915693632, −2.25487913579769268532888068223, −1.27889856704408485626853232664, −0.972818790182466361456442744907,
0.972818790182466361456442744907, 1.27889856704408485626853232664, 2.25487913579769268532888068223, 2.51249062672094151589915693632, 3.41643432170277313086471770107, 3.48616781407495673743751503909, 4.15703626568098124364510977627, 4.30824110962678800119848532316, 5.27007052034441234264207157605, 5.45215002417403430665749076969, 6.04534803929674304450213973961, 6.39227626325011928837343615527, 7.06589728763642699759212514089, 7.39560216304176002028261402091, 7.63614701002364541388417247504, 8.034776072701869713369129177743, 8.475374982140391211858934967833, 8.898230127423678495459227882582, 9.344362040845107118359055715455, 9.623023746496957437399267145123
