L(s) = 1 | + (0.5 + 0.866i)5-s + (−2.62 − 0.358i)7-s + (−2.41 + 4.18i)11-s + 2·13-s + (1.82 − 3.16i)17-s + (−2.82 − 4.89i)19-s + (−4.20 − 7.28i)23-s + (−0.499 + 0.866i)25-s + 2.17·29-s + (2.41 − 4.18i)31-s + (−1 − 2.44i)35-s + (2.82 + 4.89i)37-s − 0.171·41-s + 12.8·43-s + (0.171 + 0.297i)47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (−0.990 − 0.135i)7-s + (−0.727 + 1.26i)11-s + 0.554·13-s + (0.443 − 0.768i)17-s + (−0.648 − 1.12i)19-s + (−0.877 − 1.51i)23-s + (−0.0999 + 0.173i)25-s + 0.403·29-s + (0.433 − 0.751i)31-s + (−0.169 − 0.414i)35-s + (0.464 + 0.805i)37-s − 0.0267·41-s + 1.96·43-s + (0.0250 + 0.0433i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.327264951\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.327264951\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.62 + 0.358i)T \) |
good | 11 | \( 1 + (2.41 - 4.18i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1.82 + 3.16i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.20 + 7.28i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + (-2.41 + 4.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.171T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 + (-0.171 - 0.297i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.82 + 4.89i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.32 - 4.03i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.44 - 5.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-3.82 + 6.63i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + (8.32 + 14.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.927860894239900708334092878782, −7.984550729997755448293350798376, −7.18041928776820985954413129103, −6.56345932477489064408644240722, −5.88398197181090572024400949392, −4.75909000353120519273472846408, −4.08928388900415759221756013795, −2.80425934696759322116782140099, −2.33216782280461266691121172318, −0.53482029444019330684399405095,
0.970592377212212308577829135201, 2.29200634848773662702430083417, 3.45000419729918021257313063469, 3.91788069900409650982774646687, 5.36484403432371547884765950206, 5.93555578692099826545490245355, 6.38210506206483103841455015678, 7.72061773148124464095115178363, 8.191341045582786787174626713082, 9.011067256360728508789808257896