L(s) = 1 | + (0.707 − 0.707i)2-s + (0.866 + 0.5i)3-s − 1.00i·4-s + (0.965 − 0.258i)5-s + (0.965 − 0.258i)6-s + (−1.22 − 1.22i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.500 − 0.866i)12-s + (−0.707 + 0.707i)13-s − 1.73·14-s + (0.965 + 0.258i)15-s − 1.00·16-s + (0.366 − 0.366i)17-s + (0.965 + 0.258i)18-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (0.866 + 0.5i)3-s − 1.00i·4-s + (0.965 − 0.258i)5-s + (0.965 − 0.258i)6-s + (−1.22 − 1.22i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.500 − 0.866i)12-s + (−0.707 + 0.707i)13-s − 1.73·14-s + (0.965 + 0.258i)15-s − 1.00·16-s + (0.366 − 0.366i)17-s + (0.965 + 0.258i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.109093420\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.109093420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 13 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 47 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.93iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.745597162521543407576047493882, −9.197742165630489345237008553816, −7.917775739634194075854381472035, −6.78142645897986750357103317313, −6.26314274391038064183074770932, −4.88885904718694506623191882718, −4.42183190655996035523078919576, −3.29508093952463710283722261282, −2.69306959065556196322009056740, −1.40490270164603356212961574246,
2.23531488614914960250405145799, 2.84359757365443044089329818523, 3.59841505043628997501815685601, 5.11144187420409908001610359126, 5.87489516521264916870762739097, 6.51024226659081480062417178730, 7.15474048265387852148945429122, 8.242671449055865882205940035513, 8.795369196665696930115676170386, 9.679591992748955645206140062358