L(s) = 1 | + (2.03 + 1.96i)2-s + (5.02 − 1.30i)3-s + (0.292 + 7.99i)4-s + 5·5-s + (12.8 + 7.21i)6-s + 8.16i·7-s + (−15.0 + 16.8i)8-s + (23.6 − 13.1i)9-s + (10.1 + 9.81i)10-s − 16.0i·11-s + (11.8 + 39.8i)12-s + 41.4i·13-s + (−16.0 + 16.6i)14-s + (25.1 − 6.51i)15-s + (−63.8 + 4.67i)16-s − 83.3i·17-s + ⋯ |
L(s) = 1 | + (0.719 + 0.694i)2-s + (0.968 − 0.250i)3-s + (0.0365 + 0.999i)4-s + 0.447·5-s + (0.871 + 0.491i)6-s + 0.440i·7-s + (−0.667 + 0.744i)8-s + (0.874 − 0.485i)9-s + (0.321 + 0.310i)10-s − 0.438i·11-s + (0.286 + 0.958i)12-s + 0.884i·13-s + (−0.305 + 0.317i)14-s + (0.432 − 0.112i)15-s + (−0.997 + 0.0730i)16-s − 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.77581 + 1.69011i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.77581 + 1.69011i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.03 - 1.96i)T \) |
| 3 | \( 1 + (-5.02 + 1.30i)T \) |
| 5 | \( 1 - 5T \) |
good | 7 | \( 1 - 8.16iT - 343T^{2} \) |
| 11 | \( 1 + 16.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 41.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 83.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 54.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 11.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 245. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 14.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 485.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 70.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 488. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 886. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 780.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 520.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 296.T + 9.12e5T^{2} \) |
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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
−13.61914534864104287820340606451, −12.45622736521021980374782630112, −11.49188316497648116720247674097, −9.569864293250865005769243302458, −8.792345089312793320547862443788, −7.60064175991947861008860370358, −6.59376675787549892876799899686, −5.23890535911488812103674266153, −3.69067190161948048840272130984, −2.31875881977033604554297002870,
1.66563782669911959391888135697, 3.15115695473592181450886229770, 4.33245686562510984743337906148, 5.74824010871323510327857588660, 7.33539937453298994604256404661, 8.751740199221549349493321255086, 10.05446699254574916705167576380, 10.41442470871784673869949913198, 11.97273917200662705674114498096, 13.18419220735141604880467679837