L(s) = 1 | + (0.345 − 1.37i)2-s + (−0.707 − 0.707i)3-s + (−1.76 − 0.946i)4-s + (−0.561 + 2.16i)5-s + (−1.21 + 0.725i)6-s − 4.51·7-s + (−1.90 + 2.08i)8-s + 1.00i·9-s + (2.77 + 1.51i)10-s + (−3.44 − 3.44i)11-s + (0.576 + 1.91i)12-s + (0.113 + 0.113i)13-s + (−1.55 + 6.19i)14-s + (1.92 − 1.13i)15-s + (2.20 + 3.33i)16-s − 5.03i·17-s + ⋯ |
L(s) = 1 | + (0.244 − 0.969i)2-s + (−0.408 − 0.408i)3-s + (−0.880 − 0.473i)4-s + (−0.251 + 0.967i)5-s + (−0.495 + 0.296i)6-s − 1.70·7-s + (−0.673 + 0.738i)8-s + 0.333i·9-s + (0.877 + 0.479i)10-s + (−1.03 − 1.03i)11-s + (0.166 + 0.552i)12-s + (0.0315 + 0.0315i)13-s + (−0.416 + 1.65i)14-s + (0.497 − 0.292i)15-s + (0.551 + 0.833i)16-s − 1.22i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0820432 + 0.181926i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0820432 + 0.181926i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.345 + 1.37i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.561 - 2.16i)T \) |
good | 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 + (3.44 + 3.44i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.113 - 0.113i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.03iT - 17T^{2} \) |
| 19 | \( 1 + (0.992 - 0.992i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.00T + 23T^{2} \) |
| 29 | \( 1 + (1.01 - 1.01i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 + (1.63 - 1.63i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.35iT - 41T^{2} \) |
| 43 | \( 1 + (5.68 - 5.68i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.10iT - 47T^{2} \) |
| 53 | \( 1 + (3.27 - 3.27i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.30 + 5.30i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.87 - 5.87i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.87 - 1.87i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.635iT - 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 + (-6.39 - 6.39i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.579iT - 89T^{2} \) |
| 97 | \( 1 + 15.2iT - 97T^{2} \) |
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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
−11.40484772511433188417094098269, −10.75379261078266460272842637501, −9.924187868018854336353128346355, −8.875276757143042032791261528970, −7.33830441891278496287731582230, −6.32706277985959528426329045372, −5.28366148909343370139895619221, −3.39929404925038268261512875001, −2.78644008064039313696558650836, −0.14880834824074971559494195170,
3.42569437858897153475121106809, 4.61326439123376189619133773994, 5.56757236303365889143841303206, 6.61710508524053017631133069344, 7.65742709554653206532235553951, 8.944082400939262940858354431421, 9.577240728731433012589279834605, 10.59701006347567638676553004509, 12.31623545780633777719142578598, 12.88720528675821927403852828084