| L(s) = 1 | + (−0.110 − 0.397i)2-s + (−0.426 − 1.67i)3-s + (1.56 − 0.943i)4-s + (2.16 − 0.117i)5-s + (−0.620 + 0.355i)6-s + (0.253 + 1.54i)7-s + (−1.14 − 1.08i)8-s + (−2.63 + 1.43i)9-s + (−0.286 − 0.849i)10-s + (0.507 + 0.957i)11-s + (−2.25 − 2.22i)12-s + (−2.53 − 1.00i)13-s + (0.587 − 0.271i)14-s + (−1.12 − 3.58i)15-s + (1.40 − 2.65i)16-s + (1.49 + 0.245i)17-s + ⋯ |
| L(s) = 1 | + (−0.0781 − 0.281i)2-s + (−0.246 − 0.969i)3-s + (0.783 − 0.471i)4-s + (0.969 − 0.0525i)5-s + (−0.253 + 0.145i)6-s + (0.0958 + 0.584i)7-s + (−0.405 − 0.384i)8-s + (−0.878 + 0.477i)9-s + (−0.0904 − 0.268i)10-s + (0.153 + 0.288i)11-s + (−0.650 − 0.643i)12-s + (−0.702 − 0.279i)13-s + (0.157 − 0.0726i)14-s + (−0.289 − 0.926i)15-s + (0.351 − 0.663i)16-s + (0.363 + 0.0595i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02850 - 0.830566i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02850 - 0.830566i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.426 + 1.67i)T \) |
| 59 | \( 1 + (-3.19 - 6.98i)T \) |
| good | 2 | \( 1 + (0.110 + 0.397i)T + (-1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-2.16 + 0.117i)T + (4.97 - 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.253 - 1.54i)T + (-6.63 + 2.23i)T^{2} \) |
| 11 | \( 1 + (-0.507 - 0.957i)T + (-6.17 + 9.10i)T^{2} \) |
| 13 | \( 1 + (2.53 + 1.00i)T + (9.43 + 8.94i)T^{2} \) |
| 17 | \( 1 + (-1.49 - 0.245i)T + (16.1 + 5.42i)T^{2} \) |
| 19 | \( 1 + (2.26 - 2.66i)T + (-3.07 - 18.7i)T^{2} \) |
| 23 | \( 1 + (1.57 + 1.19i)T + (6.15 + 22.1i)T^{2} \) |
| 29 | \( 1 + (-3.75 - 1.04i)T + (24.8 + 14.9i)T^{2} \) |
| 31 | \( 1 + (4.33 - 3.68i)T + (5.01 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-5.65 - 5.96i)T + (-2.00 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-6.36 - 8.37i)T + (-10.9 + 39.5i)T^{2} \) |
| 43 | \( 1 + (8.66 + 4.59i)T + (24.1 + 35.5i)T^{2} \) |
| 47 | \( 1 + (-0.568 + 10.4i)T + (-46.7 - 5.08i)T^{2} \) |
| 53 | \( 1 + (0.757 - 2.24i)T + (-42.1 - 32.0i)T^{2} \) |
| 61 | \( 1 + (-5.58 + 1.55i)T + (52.2 - 31.4i)T^{2} \) |
| 67 | \( 1 + (-5.58 + 5.89i)T + (-3.62 - 66.9i)T^{2} \) |
| 71 | \( 1 + (12.8 + 0.697i)T + (70.5 + 7.67i)T^{2} \) |
| 73 | \( 1 + (-0.608 - 1.31i)T + (-47.2 + 55.6i)T^{2} \) |
| 79 | \( 1 + (-2.11 - 3.11i)T + (-29.2 + 73.3i)T^{2} \) |
| 83 | \( 1 + (10.1 - 2.23i)T + (75.3 - 34.8i)T^{2} \) |
| 89 | \( 1 + (-2.28 + 8.21i)T + (-76.2 - 45.8i)T^{2} \) |
| 97 | \( 1 + (0.449 - 0.971i)T + (-62.7 - 73.9i)T^{2} \) |
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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
−12.31273524890974319237900332958, −11.73139678033769821293846998956, −10.48940739004261458691408726805, −9.711139591357994920426726352684, −8.338658792757314570127180538265, −7.05296713414277285393493632968, −6.12235084415278924680786556337, −5.33840784159959590123961524113, −2.64966058184580564506517650773, −1.64438736563684596986705682502,
2.51500485495775132019677867284, 4.06160514213067848295561221502, 5.55046939793761347604488298610, 6.47662931840642328978926027780, 7.70668191291284915288786769831, 9.050085187250530114023227852605, 9.955002907482223802230167439956, 10.88727891544820139804562447509, 11.67933259871661433204609371942, 12.86861057581321366368643103833
