| L(s) = 1 | + (−2.23 − 3.29i)2-s + (−0.162 + 2.99i)3-s + (−2.89 + 7.27i)4-s + (−6.22 + 2.88i)5-s + (10.2 − 6.15i)6-s + (−3.74 + 13.4i)7-s + (−0.647 + 0.142i)8-s + (−8.94 − 0.973i)9-s + (23.3 + 14.0i)10-s + (30.5 − 28.9i)11-s + (−21.3 − 9.86i)12-s + (−24.0 + 2.61i)13-s + (52.7 − 17.7i)14-s + (−7.61 − 19.1i)15-s + (47.4 + 44.9i)16-s + (−7.41 − 26.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.789 − 1.16i)2-s + (−0.0312 + 0.576i)3-s + (−0.362 + 0.909i)4-s + (−0.556 + 0.257i)5-s + (0.695 − 0.418i)6-s + (−0.202 + 0.728i)7-s + (−0.0286 + 0.00629i)8-s + (−0.331 − 0.0360i)9-s + (0.739 + 0.444i)10-s + (0.838 − 0.794i)11-s + (−0.513 − 0.237i)12-s + (−0.513 + 0.0558i)13-s + (1.00 − 0.339i)14-s + (−0.131 − 0.329i)15-s + (0.740 + 0.701i)16-s + (−0.105 − 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.469303 - 0.617537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.469303 - 0.617537i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.162 - 2.99i)T \) |
| 59 | \( 1 + (-447. + 72.1i)T \) |
| good | 2 | \( 1 + (2.23 + 3.29i)T + (-2.96 + 7.43i)T^{2} \) |
| 5 | \( 1 + (6.22 - 2.88i)T + (80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (3.74 - 13.4i)T + (-293. - 176. i)T^{2} \) |
| 11 | \( 1 + (-30.5 + 28.9i)T + (72.0 - 1.32e3i)T^{2} \) |
| 13 | \( 1 + (24.0 - 2.61i)T + (2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (7.41 + 26.6i)T + (-4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (-123. + 94.2i)T + (1.83e3 - 6.60e3i)T^{2} \) |
| 23 | \( 1 + (-76.9 + 145. i)T + (-6.82e3 - 1.00e4i)T^{2} \) |
| 29 | \( 1 + (-66.4 + 98.0i)T + (-9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (-38.8 - 29.5i)T + (7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (279. + 61.5i)T + (4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (10.3 + 19.5i)T + (-3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (-60.6 - 57.4i)T + (4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (456. + 210. i)T + (6.72e4 + 7.91e4i)T^{2} \) |
| 53 | \( 1 + (-542. + 326. i)T + (6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (130. + 192. i)T + (-8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (-77.5 + 17.0i)T + (2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (-733. - 339. i)T + (2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-1.04e3 + 351. i)T + (3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (5.73 + 105. i)T + (-4.90e5 + 5.33e4i)T^{2} \) |
| 83 | \( 1 + (-37.5 - 229. i)T + (-5.41e5 + 1.82e5i)T^{2} \) |
| 89 | \( 1 + (442. - 652. i)T + (-2.60e5 - 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-358. - 120. i)T + (7.26e5 + 5.52e5i)T^{2} \) |
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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
−11.59527310495061663422953467440, −11.11375918734626223263689097680, −9.900874989761515030477024294220, −9.164326053849130042999626183264, −8.400584581367492314929412805085, −6.78570532186287771442590927520, −5.24483422201921121166885585479, −3.57745498603370057067885889194, −2.61432698081099346272252902138, −0.56267295240564432739757472887,
1.13431634039889094212432027078, 3.64602045709352295204820393190, 5.34212704530515596097939437423, 6.70704705408841275867093971126, 7.37182013630110680125075395651, 8.093495381808167942893778554495, 9.290843802043870671133607717148, 10.12686144586380841002065756372, 11.78028055419613634696856222557, 12.35500433005390100870935977290
