| L(s) = 1 | + (−1.49 − 0.594i)2-s + (−1.72 − 0.187i)3-s + (−1.03 − 0.978i)4-s + (−2.12 + 2.49i)5-s + (2.45 + 1.30i)6-s + (6.56 + 3.95i)7-s + (3.65 + 7.90i)8-s + (2.92 + 0.644i)9-s + (4.64 − 2.46i)10-s + (−6.49 − 0.352i)11-s + (1.59 + 1.87i)12-s + (−4.42 − 20.1i)13-s + (−7.44 − 9.79i)14-s + (4.11 − 3.90i)15-s + (−0.448 − 8.27i)16-s + (20.4 − 12.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.745 − 0.297i)2-s + (−0.573 − 0.0624i)3-s + (−0.258 − 0.244i)4-s + (−0.424 + 0.499i)5-s + (0.409 + 0.217i)6-s + (0.938 + 0.564i)7-s + (0.456 + 0.987i)8-s + (0.325 + 0.0716i)9-s + (0.464 − 0.246i)10-s + (−0.590 − 0.0320i)11-s + (0.132 + 0.156i)12-s + (−0.340 − 1.54i)13-s + (−0.532 − 0.699i)14-s + (0.274 − 0.260i)15-s + (−0.0280 − 0.517i)16-s + (1.20 − 0.722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.593246 - 0.362483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.593246 - 0.362483i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.187i)T \) |
| 59 | \( 1 + (-46.1 + 36.7i)T \) |
| good | 2 | \( 1 + (1.49 + 0.594i)T + (2.90 + 2.75i)T^{2} \) |
| 5 | \( 1 + (2.12 - 2.49i)T + (-4.04 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-6.56 - 3.95i)T + (22.9 + 43.2i)T^{2} \) |
| 11 | \( 1 + (6.49 + 0.352i)T + (120. + 13.0i)T^{2} \) |
| 13 | \( 1 + (4.42 + 20.1i)T + (-153. + 70.9i)T^{2} \) |
| 17 | \( 1 + (-20.4 + 12.2i)T + (135. - 255. i)T^{2} \) |
| 19 | \( 1 + (-5.98 + 21.5i)T + (-309. - 186. i)T^{2} \) |
| 23 | \( 1 + (2.56 - 1.73i)T + (195. - 491. i)T^{2} \) |
| 29 | \( 1 + (-5.81 - 14.5i)T + (-610. + 578. i)T^{2} \) |
| 31 | \( 1 + (-45.1 + 12.5i)T + (823. - 495. i)T^{2} \) |
| 37 | \( 1 + (-28.6 + 62.0i)T + (-886. - 1.04e3i)T^{2} \) |
| 41 | \( 1 + (4.63 - 6.83i)T + (-622. - 1.56e3i)T^{2} \) |
| 43 | \( 1 + (-16.3 + 0.885i)T + (1.83e3 - 199. i)T^{2} \) |
| 47 | \( 1 + (6.71 - 5.70i)T + (357. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + (22.7 - 42.9i)T + (-1.57e3 - 2.32e3i)T^{2} \) |
| 61 | \( 1 + (15.8 + 6.30i)T + (2.70e3 + 2.55e3i)T^{2} \) |
| 67 | \( 1 + (-23.6 - 51.1i)T + (-2.90e3 + 3.42e3i)T^{2} \) |
| 71 | \( 1 + (-19.6 - 23.1i)T + (-815. + 4.97e3i)T^{2} \) |
| 73 | \( 1 + (9.54 + 12.5i)T + (-1.42e3 + 5.13e3i)T^{2} \) |
| 79 | \( 1 + (-79.9 + 8.69i)T + (6.09e3 - 1.34e3i)T^{2} \) |
| 83 | \( 1 + (35.9 + 106. i)T + (-5.48e3 + 4.16e3i)T^{2} \) |
| 89 | \( 1 + (-150. + 60.0i)T + (5.75e3 - 5.44e3i)T^{2} \) |
| 97 | \( 1 + (108. - 143. i)T + (-2.51e3 - 9.06e3i)T^{2} \) |
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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
−11.92494233436351706274246204727, −11.12497385529096797047844374237, −10.43037311025062030011283606317, −9.423929723525201241059683084640, −8.097666689692390511506042309365, −7.49762049957792927709828769563, −5.57940433653241024194699054939, −4.94526341766061200157737991081, −2.72794419182658063923888870696, −0.73364695789938848861631787700,
1.18905304146359822055260665400, 4.04386383903266056115906047431, 4.86235987416570156931406084827, 6.54540177809497565627908113626, 7.88177164922970635519277840186, 8.212730625848015705068411836276, 9.687220863797815578114924195026, 10.41618638043459546362055767613, 11.77150838549398258675326394211, 12.32571092894508376219631732463
