L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−2.06 − 0.866i)5-s + (0.707 + 0.707i)6-s + (−0.662 + 0.662i)7-s − i·8-s − 1.00i·9-s + (0.866 − 2.06i)10-s − 3.51i·11-s + (−0.707 + 0.707i)12-s + 6.87i·13-s + (−0.662 − 0.662i)14-s + (−2.07 + 0.844i)15-s + 16-s − 2.17·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (−0.921 − 0.387i)5-s + (0.288 + 0.288i)6-s + (−0.250 + 0.250i)7-s − 0.353i·8-s − 0.333i·9-s + (0.273 − 0.651i)10-s − 1.06i·11-s + (−0.204 + 0.204i)12-s + 1.90i·13-s + (−0.176 − 0.176i)14-s + (−0.534 + 0.218i)15-s + 0.250·16-s − 0.527·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.045607019\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.045607019\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.06 + 0.866i)T \) |
| 37 | \( 1 + (1.24 - 5.95i)T \) |
good | 7 | \( 1 + (0.662 - 0.662i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.51iT - 11T^{2} \) |
| 13 | \( 1 - 6.87iT - 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + (-1.01 + 1.01i)T - 19iT^{2} \) |
| 23 | \( 1 - 9.57iT - 23T^{2} \) |
| 29 | \( 1 + (-6.49 - 6.49i)T + 29iT^{2} \) |
| 31 | \( 1 + (-2.38 + 2.38i)T - 31iT^{2} \) |
| 41 | \( 1 + 3.47iT - 41T^{2} \) |
| 43 | \( 1 - 2.74iT - 43T^{2} \) |
| 47 | \( 1 + (-3.59 + 3.59i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.93 + 2.93i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.45 + 3.45i)T - 59iT^{2} \) |
| 61 | \( 1 + (9.99 - 9.99i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.14 - 1.14i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.32T + 71T^{2} \) |
| 73 | \( 1 + (-0.292 + 0.292i)T - 73iT^{2} \) |
| 79 | \( 1 + (7.95 - 7.95i)T - 79iT^{2} \) |
| 83 | \( 1 + (1.46 + 1.46i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.43 - 6.43i)T + 89iT^{2} \) |
| 97 | \( 1 - 0.301T + 97T^{2} \) |
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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
−9.697289876600750144192477220216, −8.868782381081846133019088325575, −8.590843948584604167722098345798, −7.52386771922060004108003299245, −6.90427530307665853553135756059, −6.06661984320949920738855331664, −4.91997743313898048046457872418, −3.99785469938828077716273990989, −3.06982097827231359825278196762, −1.33149023307179151271119857762,
0.47943109858464493491704585173, 2.47530322000802822706971579008, 3.17523998640869354335886748781, 4.24049707860916864655268664119, 4.82278051113974902680505434159, 6.23579358401692105812774823277, 7.35019619425989975341409922584, 8.086857853525915565989084397577, 8.733586925177304057311526668908, 9.948753608156544343512298927931