L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.258 + 0.965i)3-s + (−0.499 + 0.866i)4-s + (−0.707 + 0.707i)5-s + (0.965 − 0.258i)6-s + (0.866 + 0.5i)7-s + 0.999·8-s + (0.965 + 0.258i)10-s + (−0.707 − 0.707i)12-s + (0.965 − 0.258i)13-s − 0.999i·14-s + (−0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)20-s + (−0.707 + 0.707i)21-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.258 + 0.965i)3-s + (−0.499 + 0.866i)4-s + (−0.707 + 0.707i)5-s + (0.965 − 0.258i)6-s + (0.866 + 0.5i)7-s + 0.999·8-s + (0.965 + 0.258i)10-s + (−0.707 − 0.707i)12-s + (0.965 − 0.258i)13-s − 0.999i·14-s + (−0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)20-s + (−0.707 + 0.707i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7504235997\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7504235997\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.965 + 0.258i)T \) |
good | 3 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−10.21029908432605189222905212906, −9.017766555087418868707888470150, −8.574319245102507579698163012108, −7.67317330689371488351086042722, −6.84344515530269341220323411762, −5.36515401504113884178485777011, −4.64192752924750129547351475370, −3.72451709153063195295504893179, −3.02205662207363167523471711706, −1.59742167828208374735397518248,
0.869465272053156630186402336908, 1.70160463616609261015266837371, 3.89363635928299060705027660572, 4.66597823815005745002130749084, 5.59301474793750099341923291104, 6.57759040377835981780997928259, 7.17802737594899444043711388117, 8.077618013566943434574470542355, 8.288520229789178202395298314397, 9.301038220664391928320067466402