| L(s) = 1 | + (−1.23 + 3.67i)2-s + (2.23 − 1.99i)3-s + (−8.78 − 6.67i)4-s + (2.27 + 0.906i)5-s + (4.57 + 10.6i)6-s + (0.939 − 0.434i)7-s + (22.5 − 15.3i)8-s + (1.01 − 8.94i)9-s + (−6.14 + 7.23i)10-s + (17.5 + 4.86i)11-s + (−32.9 + 2.60i)12-s + (−4.59 + 8.66i)13-s + (0.434 + 3.99i)14-s + (6.90 − 2.51i)15-s + (16.4 + 59.3i)16-s + (−4.21 + 9.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.618 + 1.83i)2-s + (0.745 − 0.666i)3-s + (−2.19 − 1.66i)4-s + (0.454 + 0.181i)5-s + (0.761 + 1.78i)6-s + (0.134 − 0.0621i)7-s + (2.82 − 1.91i)8-s + (0.112 − 0.993i)9-s + (−0.614 + 0.723i)10-s + (1.59 + 0.442i)11-s + (−2.74 + 0.217i)12-s + (−0.353 + 0.666i)13-s + (0.0310 + 0.285i)14-s + (0.460 − 0.167i)15-s + (1.02 + 3.70i)16-s + (−0.247 + 0.535i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0986 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0986 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05147 + 0.952360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05147 + 0.952360i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.23 + 1.99i)T \) |
| 59 | \( 1 + (4.85 - 58.8i)T \) |
| good | 2 | \( 1 + (1.23 - 3.67i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (-2.27 - 0.906i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 0.434i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-17.5 - 4.86i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (4.59 - 8.66i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (4.21 - 9.10i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-30.5 + 6.72i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-13.2 - 2.17i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-6.34 - 18.8i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (12.7 + 2.81i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-3.23 + 4.76i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (13.4 - 2.21i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (17.8 + 64.3i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-36.7 + 14.6i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-33.8 + 28.7i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-100. - 33.7i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (12.7 + 18.8i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (127. - 50.9i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (15.7 - 1.71i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (89.8 - 54.0i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-7.51 + 0.407i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (28.5 + 84.7i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (123. + 13.4i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−13.29630972197778149746450059413, −11.88236432319583138560287978566, −9.963812177677081016500735116248, −9.226251911107377626678621746952, −8.573968841905484162442440030210, −7.17711622288279393681363871518, −6.90348970752429077882936127407, −5.69956340052185236284424239017, −4.13096918163068457939951639851, −1.39245910129935744326962709107,
1.36996433338175928671344715879, 2.88230611721971167743774192302, 3.84555394066214996224144362403, 5.12833354031394553488064328775, 7.67848991962765452640685143784, 8.785148259478299119742828380744, 9.481920741825723114336941276492, 10.00892652464825100411755896584, 11.24864920338985262540715464599, 11.84857593713127836294739114896
