L(s) = 1 | + (1 − 1.41i)3-s + (−1.00 − 2.82i)9-s + 2.82i·11-s + 5.65i·17-s − 2·19-s − 5·25-s + (−5.00 − 1.41i)27-s + (4.00 + 2.82i)33-s − 11.3i·41-s + 10·43-s + 7·49-s + (8.00 + 5.65i)51-s + (−2 + 2.82i)57-s − 14.1i·59-s − 14·67-s + ⋯ |
L(s) = 1 | + (0.577 − 0.816i)3-s + (−0.333 − 0.942i)9-s + 0.852i·11-s + 1.37i·17-s − 0.458·19-s − 25-s + (−0.962 − 0.272i)27-s + (0.696 + 0.492i)33-s − 1.76i·41-s + 1.52·43-s + 49-s + (1.12 + 0.792i)51-s + (−0.264 + 0.374i)57-s − 1.84i·59-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08458 - 0.344722i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08458 - 0.344722i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 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
−13.83052280484096635574528108503, −12.77691477777785881349281164349, −12.10706443513036903495494494042, −10.63625573446747533218639610596, −9.349238772081262739397023984498, −8.221852677201233064086912461177, −7.18846721052102894550229109732, −5.94979681928227901041780787872, −3.94892189051276045666934919575, −2.05544024252426502892033942734,
2.85796536911498613400902789739, 4.35093785569216056385832285131, 5.77683474317835016152986578228, 7.55043553750843323511328883538, 8.727473141562628758708798405880, 9.656089035790711538682577334756, 10.79247496690339236535851258895, 11.76916993795214137709269398896, 13.36478006518469625335286818462, 14.06394328773157206212895907659