L(s) = 1 | + (−0.965 + 0.258i)3-s + (0.5 + 0.866i)4-s + (0.965 + 1.67i)7-s + (0.866 − 0.499i)9-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.499 + 0.866i)16-s + (0.133 − 0.5i)19-s + (−1.36 − 1.36i)21-s + (−0.707 + 0.707i)27-s + (−0.965 + 1.67i)28-s + (−1 + i)31-s + (0.866 + 0.5i)36-s + (0.707 − 1.22i)37-s + (0.866 + 0.500i)39-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)3-s + (0.5 + 0.866i)4-s + (0.965 + 1.67i)7-s + (0.866 − 0.499i)9-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.499 + 0.866i)16-s + (0.133 − 0.5i)19-s + (−1.36 − 1.36i)21-s + (−0.707 + 0.707i)27-s + (−0.965 + 1.67i)28-s + (−1 + i)31-s + (0.866 + 0.5i)36-s + (0.707 − 1.22i)37-s + (0.866 + 0.500i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8882996162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8882996162\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 37 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + 0.517iT - T^{2} \) |
| 79 | \( 1 + 1.73iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
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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
−10.81816982999094923778048672683, −9.494901505879654070588031572815, −8.778293058077922621474412170332, −7.84394170868152666664248234795, −7.09812233274576701276015421594, −5.94983018709411848850862996572, −5.32264856220692344149605842177, −4.43240964200390658385865804437, −3.02520461936076996854922964840, −1.98066638880397077597535979469,
1.02137752544723380666348283364, 2.01692626198572854955352423430, 4.08986563174561049671163148132, 4.79469797197852042278808615165, 5.67589306032189978982662107493, 6.70418147902140100516070189872, 7.27079679632430611871617541209, 7.957577604566815718112111915348, 9.604771180728428468212866752284, 10.15038757406011153761720966346