| L(s) = 1 | + (0.456 − 2.18i)5-s − 4.37i·7-s + 5.58i·11-s − 4.37·13-s + 5.58i·17-s + 4i·19-s + (−4.58 − 1.99i)25-s + 2.55i·29-s − 5.29·31-s + (−9.58 − 1.99i)35-s − 2.55·37-s − 6·41-s + 11.1·43-s + 6.92i·47-s − 12.1·49-s + ⋯ |
| L(s) = 1 | + (0.204 − 0.978i)5-s − 1.65i·7-s + 1.68i·11-s − 1.21·13-s + 1.35i·17-s + 0.917i·19-s + (−0.916 − 0.399i)25-s + 0.473i·29-s − 0.950·31-s + (−1.61 − 0.338i)35-s − 0.419·37-s − 0.937·41-s + 1.70·43-s + 1.01i·47-s − 1.73·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0560 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0560 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7085825748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7085825748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.456 + 2.18i)T \) |
| good | 7 | \( 1 + 4.37iT - 7T^{2} \) |
| 11 | \( 1 - 5.58iT - 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 5.58iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2.55iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 + 2.55T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 + 7.84T + 53T^{2} \) |
| 59 | \( 1 - 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−9.117348725003818003392455571300, −7.973714500077339153512479320170, −7.53925680736965808674693371552, −6.89739589563113273173447948821, −5.86659391158872966247740729712, −4.83621200841195328431152131150, −4.37447536658433905922674626712, −3.65224351498962602155126293182, −2.03378251126436996687616232777, −1.31530181761542214120275817810,
0.21767512147421907868713069862, 2.21591754775059521836641798102, 2.75128024370606693765584068167, 3.44780384459930987911864661280, 4.97876371689175509588879773054, 5.51219064496146720254993744484, 6.24108518965244920656427101419, 7.01337531379785734322032413942, 7.80066275492025121276729517637, 8.733623071889714017006858961069
