| L(s) = 1 | + (1.38 + 0.261i)2-s + (1.70 − 0.328i)3-s + (1.86 + 0.727i)4-s + (0.232 − 0.264i)5-s + (2.44 − 0.0112i)6-s + (−2.67 + 0.352i)7-s + (2.39 + 1.49i)8-s + (2.78 − 1.11i)9-s + (0.391 − 0.307i)10-s + (1.81 − 3.67i)11-s + (3.40 + 0.625i)12-s + (0.566 − 1.66i)13-s + (−3.81 − 0.211i)14-s + (0.307 − 0.526i)15-s + (2.94 + 2.71i)16-s + (−4.32 + 4.32i)17-s + ⋯ |
| L(s) = 1 | + (0.982 + 0.185i)2-s + (0.981 − 0.189i)3-s + (0.931 + 0.363i)4-s + (0.103 − 0.118i)5-s + (0.999 − 0.00458i)6-s + (−1.01 + 0.133i)7-s + (0.847 + 0.530i)8-s + (0.928 − 0.372i)9-s + (0.123 − 0.0970i)10-s + (0.546 − 1.10i)11-s + (0.983 + 0.180i)12-s + (0.157 − 0.463i)13-s + (−1.01 − 0.0564i)14-s + (0.0794 − 0.135i)15-s + (0.735 + 0.678i)16-s + (−1.04 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.45481 + 0.104350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.45481 + 0.104350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.261i)T \) |
| 3 | \( 1 + (-1.70 + 0.328i)T \) |
| good | 5 | \( 1 + (-0.232 + 0.264i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (2.67 - 0.352i)T + (6.76 - 1.81i)T^{2} \) |
| 11 | \( 1 + (-1.81 + 3.67i)T + (-6.69 - 8.72i)T^{2} \) |
| 13 | \( 1 + (-0.566 + 1.66i)T + (-10.3 - 7.91i)T^{2} \) |
| 17 | \( 1 + (4.32 - 4.32i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.761 + 3.82i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (0.977 - 7.42i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (3.98 + 0.261i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (-2.59 - 1.49i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.86 - 9.36i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.21 + 9.20i)T + (-39.6 - 10.6i)T^{2} \) |
| 43 | \( 1 + (9.03 + 4.45i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (7.40 + 1.98i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.62 + 9.90i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-3.40 + 3.87i)T + (-7.70 - 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.0222 + 0.338i)T + (-60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (-1.22 + 0.606i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (-3.87 + 9.34i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.45 - 5.92i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (1.82 - 6.81i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-11.0 + 9.69i)T + (10.8 - 82.2i)T^{2} \) |
| 89 | \( 1 + (-12.8 - 5.33i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (10.8 - 6.24i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.89408335612634924195409577355, −9.761446304419784228546501745054, −8.827589563725433577997236052877, −8.080748692793389621579616492240, −6.85180911524925999953262826787, −6.33779320568865151074325745555, −5.14829881086224882225264881189, −3.60963886289576675938540799366, −3.33199774244933461716925875981, −1.83858602395856756202824830447,
1.98122929917904263111007649870, 2.91394524449026085752066851585, 4.08974496163280627022986512691, 4.66906239344903059985829173470, 6.42147524047556151327011996796, 6.79687763232683999809272203318, 7.920493923146765060333869466230, 9.239817140948759144815993643244, 9.838919729836082455853721860821, 10.62179034319042945769746558129
