| L(s) = 1 | + (−2.13 + 0.951i)2-s + (−0.620 + 1.61i)3-s + (2.32 − 2.58i)4-s + (−1.69 + 1.46i)5-s + (−0.212 − 4.04i)6-s + (−0.915 + 1.58i)7-s + (−1.06 + 3.27i)8-s + (−2.23 − 2.00i)9-s + (2.22 − 4.73i)10-s + (−1.75 + 0.780i)11-s + (2.73 + 5.35i)12-s + (−0.319 − 0.142i)13-s + (0.447 − 4.26i)14-s + (−1.31 − 3.64i)15-s + (−0.116 − 1.10i)16-s + (−0.815 + 2.50i)17-s + ⋯ |
| L(s) = 1 | + (−1.51 + 0.672i)2-s + (−0.358 + 0.933i)3-s + (1.16 − 1.29i)4-s + (−0.757 + 0.653i)5-s + (−0.0868 − 1.65i)6-s + (−0.346 + 0.599i)7-s + (−0.376 + 1.15i)8-s + (−0.743 − 0.668i)9-s + (0.704 − 1.49i)10-s + (−0.528 + 0.235i)11-s + (0.788 + 1.54i)12-s + (−0.0886 − 0.0394i)13-s + (0.119 − 1.13i)14-s + (−0.338 − 0.940i)15-s + (−0.0290 − 0.276i)16-s + (−0.197 + 0.608i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0275 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0275 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0546026 - 0.0561286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0546026 - 0.0561286i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.620 - 1.61i)T \) |
| 5 | \( 1 + (1.69 - 1.46i)T \) |
| good | 2 | \( 1 + (2.13 - 0.951i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (0.915 - 1.58i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.75 - 0.780i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (0.319 + 0.142i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (0.815 - 2.50i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.57 + 4.86i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.789 + 7.51i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-7.09 - 1.50i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (6.88 - 1.46i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (4.22 - 3.06i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.49 + 1.11i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (3.95 - 6.85i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.9 + 2.33i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.727 - 2.23i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (5.30 + 2.36i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (7.48 - 3.33i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-4.64 + 0.988i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-4.20 - 12.9i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (10.4 + 7.59i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (6.36 + 1.35i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (6.26 + 6.95i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (8.55 + 6.21i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-13.4 - 2.86i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.65417884615983646724484912800, −11.49772164576729654348159215930, −10.63530501466060473523462960947, −10.07583849839719074969491899090, −8.940747776411889664150912573298, −8.311298681752260806122357250702, −7.01375970449552314347411988871, −6.22800234634244685549742737963, −4.73128418648186977793345475013, −2.96439019032410865764760371009,
0.11085257464336610470702780773, 1.53438172106057577291283855543, 3.32016011008805099416032120495, 5.31962772929197741099617642766, 7.04277227262567838134771403581, 7.73043103137799927772657807524, 8.464397339858688107861337555098, 9.569179348757967624260874690294, 10.60565108554507053898090686813, 11.51234218042482629602345564793
