L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 1.41i·5-s + 7-s + 2.82i·8-s + 2.00·10-s − 2.82i·11-s − 4.24i·13-s − 1.41i·14-s + 4.00·16-s + 6·17-s − 4.24i·19-s − 2.82i·20-s − 4.00·22-s + 6·23-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 0.632i·5-s + 0.377·7-s + 1.00i·8-s + 0.632·10-s − 0.852i·11-s − 1.17i·13-s − 0.377i·14-s + 1.00·16-s + 1.45·17-s − 0.973i·19-s − 0.632i·20-s − 0.852·22-s + 1.25·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.951411 - 0.951411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.951411 - 0.951411i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 1.41iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.69663240111516261178934131882, −10.16349232710368384616591770146, −9.002321149579703297555318913908, −8.213700238104133155937845540303, −7.21235993344681281482262024124, −5.73897235843596944810450150580, −4.94486771904310955657934264019, −3.41706763986660724589316201429, −2.80761367022671508585790385587, −0.978967310572132702839073127709,
1.44664328594197053453024460775, 3.65943890560118029816277891516, 4.77505236191553178206315729560, 5.42271579476445323548618189862, 6.68407790506238318675937481741, 7.48552588621373163826546283391, 8.360725636778285381215973336333, 9.253161678510047364510449930347, 9.877216842551701482527123592479, 11.11437400160768083366740731815