L(s) = 1 | + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)4-s + (0.965 − 0.258i)7-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)12-s + (−0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (−0.866 − 1.5i)19-s + (−0.499 − 0.866i)21-s + (0.707 + 0.707i)27-s + (0.965 + 0.258i)28-s − 36-s + (−1.67 − 0.448i)37-s + (0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (0.866 − 0.499i)49-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)4-s + (0.965 − 0.258i)7-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)12-s + (−0.707 + 0.707i)13-s + (0.499 + 0.866i)16-s + (−0.866 − 1.5i)19-s + (−0.499 − 0.866i)21-s + (0.707 + 0.707i)27-s + (0.965 + 0.258i)28-s − 36-s + (−1.67 − 0.448i)37-s + (0.866 + 0.500i)39-s + (0.707 − 0.707i)48-s + (0.866 − 0.499i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.004759463\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004759463\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.965 + 0.258i)T \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
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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
−11.20850773412826818690495518803, −10.51636380611249884633618555736, −8.911157681126444530990841185717, −8.159727468546776311303523826892, −7.12622957452280402084405631376, −6.86617906086763478105011728454, −5.54458117085940507784996483608, −4.37837114333564791038553300389, −2.68599220391535101980008274751, −1.75017213250817479610146388580,
1.93422821101657829483960274435, 3.31966094801932222018327337308, 4.71787999659229431119451120672, 5.49874314575324618887090186657, 6.33016992215885247529100964567, 7.64476829139186849407586067741, 8.479199127412655074424145371704, 9.646041102116797199558271163784, 10.49853657569453420316314641092, 10.85265194667458318473597460308