L(s) = 1 | + (0.763 − 0.763i)5-s − 1.33·7-s + (−1.95 − 1.95i)11-s + (−4.18 + 4.18i)13-s + 4.03i·17-s + (−4.26 − 4.26i)19-s + 8.86i·23-s + 3.83i·25-s + (1.23 + 1.23i)29-s − 2.87i·31-s + (−1.02 + 1.02i)35-s + (−0.434 − 0.434i)37-s + 7.81·41-s + (−5.49 + 5.49i)43-s − 3.20·47-s + ⋯ |
L(s) = 1 | + (0.341 − 0.341i)5-s − 0.505·7-s + (−0.590 − 0.590i)11-s + (−1.16 + 1.16i)13-s + 0.978i·17-s + (−0.979 − 0.979i)19-s + 1.84i·23-s + 0.766i·25-s + (0.230 + 0.230i)29-s − 0.516i·31-s + (−0.172 + 0.172i)35-s + (−0.0714 − 0.0714i)37-s + 1.21·41-s + (−0.838 + 0.838i)43-s − 0.467·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5785465493\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5785465493\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.763 + 0.763i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + (1.95 + 1.95i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.18 - 4.18i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.03iT - 17T^{2} \) |
| 19 | \( 1 + (4.26 + 4.26i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.86iT - 23T^{2} \) |
| 29 | \( 1 + (-1.23 - 1.23i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.87iT - 31T^{2} \) |
| 37 | \( 1 + (0.434 + 0.434i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 43 | \( 1 + (5.49 - 5.49i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 + (-4.06 + 4.06i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.71 - 4.71i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.26 - 3.26i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.44 + 5.44i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.76iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 + (9.73 - 9.73i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.864686102145832343621082076707, −9.382378013401124241265950444865, −8.576546797336235452584544070606, −7.58365691138225985328433160885, −6.75975010797860538157034957097, −5.86656479312152912737882554940, −5.00491747680908749722759325571, −4.01648424919913170631923825865, −2.82966494234628033509765243309, −1.68794411594019853430732158063,
0.23196881653840357469760735910, 2.31121245967201266160076259395, 2.92238332407387129417875533483, 4.37107781746088551933890013044, 5.19865485367771628547693314474, 6.20991472195774924048051140533, 6.96791131532960356559980067774, 7.84683618833459127085557003475, 8.618943144879887729575727552848, 9.838540563858841296180598102338