L(s) = 1 | − 1.30·3-s + 4-s − 1.91·5-s + 0.715·9-s − 1.30·12-s + 2.51·15-s + 16-s − 1.91·20-s + 0.830·23-s + 2.68·25-s + 0.372·27-s + 0.830·31-s + 0.715·36-s − 0.284·37-s − 1.37·45-s − 0.284·47-s − 1.30·48-s + 49-s + 1.68·53-s + 1.68·59-s + 2.51·60-s + 64-s − 1.91·67-s − 1.08·69-s − 0.284·71-s − 3.51·75-s − 1.91·80-s + ⋯ |
L(s) = 1 | − 1.30·3-s + 4-s − 1.91·5-s + 0.715·9-s − 1.30·12-s + 2.51·15-s + 16-s − 1.91·20-s + 0.830·23-s + 2.68·25-s + 0.372·27-s + 0.830·31-s + 0.715·36-s − 0.284·37-s − 1.37·45-s − 0.284·47-s − 1.30·48-s + 49-s + 1.68·53-s + 1.68·59-s + 2.51·60-s + 64-s − 1.91·67-s − 1.08·69-s − 0.284·71-s − 3.51·75-s − 1.91·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6054502917\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6054502917\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 1.30T + T^{2} \) |
| 5 | \( 1 + 1.91T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.830T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.830T + T^{2} \) |
| 37 | \( 1 + 0.284T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.284T + T^{2} \) |
| 53 | \( 1 - 1.68T + T^{2} \) |
| 59 | \( 1 - 1.68T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.91T + T^{2} \) |
| 71 | \( 1 + 0.284T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.68T + T^{2} \) |
| 97 | \( 1 - 0.830T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36104659449294107739410569663, −8.812721993111486638125499021451, −7.989096494861037211424716082526, −7.16098710873386051791740572478, −6.73620236272132343518057220286, −5.69316180881577233681750332230, −4.79127628781707694642918116446, −3.84626408250687038106870054488, −2.81768355897143033091156001504, −0.899034298214157352749991195487,
0.899034298214157352749991195487, 2.81768355897143033091156001504, 3.84626408250687038106870054488, 4.79127628781707694642918116446, 5.69316180881577233681750332230, 6.73620236272132343518057220286, 7.16098710873386051791740572478, 7.989096494861037211424716082526, 8.812721993111486638125499021451, 10.36104659449294107739410569663