| L(s) = 1 | + 12·3-s + 4·4-s + 16·5-s + 90·9-s + 48·12-s + 192·15-s + 12·16-s + 64·20-s + 60·25-s + 540·27-s + 360·36-s − 160·37-s + 1.44e3·45-s + 152·47-s + 144·48-s − 80·49-s + 128·59-s + 768·60-s + 32·64-s + 720·75-s + 192·80-s + 2.83e3·81-s + 192·89-s + 240·100-s + 2.16e3·108-s − 1.92e3·111-s − 114·121-s + ⋯ |
| L(s) = 1 | + 4·3-s + 4-s + 16/5·5-s + 10·9-s + 4·12-s + 64/5·15-s + 3/4·16-s + 16/5·20-s + 12/5·25-s + 20·27-s + 10·36-s − 4.32·37-s + 32·45-s + 3.23·47-s + 3·48-s − 1.63·49-s + 2.16·59-s + 64/5·60-s + 1/2·64-s + 48/5·75-s + 12/5·80-s + 35·81-s + 2.15·89-s + 12/5·100-s + 20·108-s − 17.2·111-s − 0.942·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(77.96827211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(77.96827211\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 7 | $C_2^2$ | \( 1 + 80 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 114 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 306 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 400 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 624 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 702 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1520 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 498 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 752 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 4194 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 4850 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 4386 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7774 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 12304 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10930 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 9918 T^{2} + p^{4} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73129994074391101940574706497, −7.51862200143136325074218456220, −7.27604459567666820927537818522, −7.16291731258007014011124756056, −7.06474827586319750877451264199, −6.50559538879379130291697890289, −6.41760291851600836009739551460, −6.10690406890972254785645324373, −6.00710083504488931632037827263, −5.37779195389710314172230951170, −5.27692201291022549034504686873, −4.87275489013279730208218267307, −4.85470694246399179732455071684, −3.91936991235741307856299191226, −3.85509554341841684321301850025, −3.75453338438451728133325966966, −3.53626558500447310075123969054, −2.82453763010823610352695875273, −2.77550800112476519278113730663, −2.34028896565438572491294082504, −2.31542003875817134947101201388, −1.80539213925927309438310021876, −1.78476526577488088343669920799, −1.56579466399330709624426553961, −1.04438681022381546266929359607,
1.04438681022381546266929359607, 1.56579466399330709624426553961, 1.78476526577488088343669920799, 1.80539213925927309438310021876, 2.31542003875817134947101201388, 2.34028896565438572491294082504, 2.77550800112476519278113730663, 2.82453763010823610352695875273, 3.53626558500447310075123969054, 3.75453338438451728133325966966, 3.85509554341841684321301850025, 3.91936991235741307856299191226, 4.85470694246399179732455071684, 4.87275489013279730208218267307, 5.27692201291022549034504686873, 5.37779195389710314172230951170, 6.00710083504488931632037827263, 6.10690406890972254785645324373, 6.41760291851600836009739551460, 6.50559538879379130291697890289, 7.06474827586319750877451264199, 7.16291731258007014011124756056, 7.27604459567666820927537818522, 7.51862200143136325074218456220, 7.73129994074391101940574706497
